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GIFT  OF 


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, 


THE  ELEMENTS  OF 

PLANE   AND   SPHERICAL 
TRIGONOMETRY 


BY 
THOMAS   U.  TAYLOR,  C.E.,  M.C.E. 

Professor  of  Applied  Mathematics  in  the  University 
of  Texas,  Austin,  Texas 


CHARLES   PURYEAR,  M.A.,  C.E. 

Professor  of  Mathematics  in  the  Agricultural  and 

Mechanical  College  of  Texas,  College 

Station,  Texas 


GINN  AND  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON" 
ATLANTA  •  DALLAS  •  COLUMBUS  •  SAN  FRANCISCO 


Entered  at  Stationers'  Hall 


Copyright,  1902,  by 
Thomas  U.  Taylor  and  Charles  Puryear 


ALL  RIGHTS  RESERVED 
A  318.10 


<  • 


G£B^ 


G1NN   AND  COMPANY-  PRO- 
PRIETORS ■  BOSTON"  •  U.S.A. 


'/. 


// 


/ 


PREFATORY  NOTE 


The  authors  have  endeavored  to  present  the  essential  parts 
of  trigonometry  in  a  form  adapted  to  practical  applications. 
They  have  had  in  mind  particularly  the  requirements  of 
schools  of  technology.  In  high-school  courses  it  is  recom- 
mended that  the  chapters  and  sections  marked  with  a  star 
be  omitted. 

Most  of  the  problems  and  numerical  exercises  are  new. 
The  answers  have  been  purposely  omitted  in  some  cases  in 
the  solution  of  triangles,  in  the  belief  that  the  student 
should  be  given  considerable  .practice  in  the  use  of  the 
check  formulas. 

In  the  preparation  of  the  book  the  authors  have  freely 
consulted  standard  works  on  the  subject.  They  are  also 
indebted  to  Dr.  H.  Y.  Benedict  of  the  University  of  Texas, 
and  to  Dr.  W.  H.  Bruce  of  the  Denton  Normal  Institute, 
Denton,  Texas,  for  many  valuable  suggestions. 


THOMAS   U.   TAYLOR, 
CHARLES   PUR YEAR. 


October,  1902. 


111 


457955 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/elementsofplanesOOtaylrich 


CONTENTS 


CHAPTER  PAGE 

I.     Trigonometric  Functions  of  an  Acute  Angle      .         .  1 

II.     Logarithms        .........  10 

III.  Solution  of  Right  Triangles 20 

IV.  Trigonometric  Functions  of  ant  Angle        ...  29 

V.     Addition  and  Subtraction  Formulas     ....  48 

VI.     Properties  of  Triangles  ;   Solution  of  Oblique  Tri- 
angles       61 

VII.     Geometrical  Applications 81 

VIII.     Radian  Measure  ;   Inverse  Functions    ....  91 

IX.     Surveying 104 

X.     Spherical  Trigonometry          .         .         .         .         .         .  116 

APPENDIX 

Explanation  of  Tables    .......  148 

Tables 


^*=  Era  \  ®  ******   ^        '^A^A; 


T 


j  -  & 


PLANE  TRIGONOMETRY 


CHAPTER   I 


TRIGONOMETRIC   FUNCTIONS   OF   AN   ACUTE  ANGLE 


1.  Definition.  The  six  parts  of  a  triangle  are  the  three 
sides  and  the  three  angles.  In  a  plane  triangle,  if  any  three 
parts  be  given,  provided  at  least  one  of  them  be  a  side,  the 
remaining  parts  can  be  computed.  Finding  the  unknown 
parts  is  called  solving  the  triangle. 

Plane  trigonometry  is,  primarily,  that  branch  of  mathe- 
matics which  has  for  its  object  the  solution  of  plane  tri- 
angles. In  the  extended  sense  in  which  the  term  is  now 
used,  it  includes  also  the  investigation  of  the  relations  of 
angular  magnitudes  in  general. 

2.  Functions.  Suppose  y  =  5x2  —  2  cc  +  3.  Then  it  is 
evident  that  the  value  of  y  depends  upon  that  of  x.  Thus, 
if  x  =*  3,  y  =  45  -  6  +  3  =  42  ;  if  x  =  1,  y  =  6  ;  if  x  =  0, 
y  =  3.  When  one  quantity  is  so  connected  with  another  that 
the  value  of  the  one  depends  upon  the  value  of  the  other, 
the  one  is  said  to  be  a  function  of  the  other.  In  the  above 
equation,  y  is  a  function  of  x. 

3.  Trigonometric  Functions  of  an 
Acute  Angle.  Let  MAN  (Fig.  1) 
be  any  acute  angle.  From  any 
point,  B,  in  one  of  the  sides  draw 
BC  perpendioular  to  the  other  side, 
forming  the  right  triangle  A  CB.    Let  Z  MAN  =  A 

1 


Fig.  1 


By  taking 


2  PLANE    TRIGONOMETRY 

the  sides  of  this  triangle  two  and  two,  we  can  form  the  six  ratios 

BC    AC    BC    AC    AB    AB      m.  -  „  ,,  ..       , 

,  ,  — 5  f  9 The  values  oi  these  ratios  do 

AB    AB    AC    BC    AC    BC 

not  depend  upon  the  lengths  of  the  sides  of  the  triangle  in 

which  they  occur;   for,  if  any  other  point  in  AN,  as  B',  be 

taken,  the  triangle  AC'B'  is  similar  to  the  triangle  ACB,  and 

B'C'      BC    AC'      AC 
we  have  — — -  =  — -  >  -— ;  =  — —  >  etc.,  since  homologous  sides 
AB'       AB    AB'       AB 

of  similar  triangles  are  proportional.     But  if  A  be  made  to 

vary,  these  ratios  will  change   in  value,   since  the  triangles 

corresponding  to  different  values  of  A  are  not  similar  to  the 

triangle  ACB  considered   above.      These   six  ratios  depend, 

therefore,  for  their  values  upon  the  value  of  A.      They  are, 

accordingly,  functions  of  A   and  are  called  the  trigonometric 

functions  of  the  angle  A. 

The  trigonometric  functions  form  an  important  class  of 
functions,  and  an  exhaustive  study  of  them  is  essential  in 
higher  mathematics. 

4.  Names  of  the  Trigonometric  Functions.  Let  A  CB  (Fig.  2) 
be  a  right  triangle.  The  trigonometric  functions  of  either  of 
its  acute  angles  are  named  as  follows : 

The  sine  is  the  ratio  of  the  opposite 

side  to  the  hypotenuse ; 
The  cosine  is  the  ratio  of  the  adjacent 

side  to  the  hypotenuse ; 
The  tangent  is  the  ratio  of  the  oppo- 
site side  to  the  adjacent  side  ; 
The  cotangent  is  the  ratio  of  the  adjacent  side  to  the  oppo- 
site side ; 
The  secant  is  the  ratio  of  the  hypotenuse  to  the  adjacent 

side; 
The  cosecant  is  the  ratio  of  the  hypotenuse  to  the  opposite 
side. 


TRIGONOMETRIC    FUNCTIONS 


Applying  these  definitions  to  the  angle  A,  we  have 


sine 

of  the 

angle 

A 

BC 

AC 

cosine 

a 

it 

"a£> 

tangent 

ii 

a 

AC' 

cotangent 

a 

u 

AC 
BC' 

secant 

tt 

a 

AB 

it    ; 

AC* 

cosecant 

ti 

a 

«45. 

BC 

"  Sine  of  the  angle  A,"  "  cosine  of  the  angle  A  "  are  usually 
written  sin  A,  cos  A;  similar  abbreviations  being  used  for  the 
other  functions. 

Then,  denoting  the  sides  of  the  triangle  by  a,  b,  c,  as  indi- 
cated in  the  figure,  the  definitions  for  the  angle  A  take  the  form 


sin  A  = 


cot  A  =  -  > 
U 


cos  A  =  -  ? 
c 


sec  A  =  - 1 
b 


a  .       c 

tan  A  =  -  ?  esc  A  =  -  • 

b  a 

It  should  be  noted  that  each  of  these  functions,  being  the 
ratio  of  the  length  of  one  line  to  that  of  another,  is  a  number. 

Questions.  Is  the  sine  of  A  greater  or  less  than  1  ?  How  is  it  with  each  of 
the  other  functions  ? 

Two  other  functions  are  defined  as  follows  :  The  versed  sine 
of  an  angle  equals  unity  minus  its  cosine;  the  coversed  sine  is 
unity  minus  the  sine.     Thus, 

-i       b  •      a       1       a 

1 ?  coversm  A  =  1  —  -  • 

c  c 


versm  A 


sin  B 

_b 
c 

cos  B 

a 

c 

tan  B 

_  b 
a 

cot  B 

a 

PLANE   TRIGONOMETRY 

Applying  the  above  definitions  to  the  angle  B,  we  have 

c 
sec  B  —  - ) 
a 

_       c 

CSC  £  =  7  > 

a 

versm  i?  =  1 > 

c 

coversm  B  =  1 

c 

The  angles  ^4  and  B  are  complements  of  each  other,  and  by 
comparing  their  functions  we  see  that 

sin  A  =  cos  B,  cos  A  =  sin  Z?,  etc. 

That  is,  the  sine  of  an  angle  equals  the  cosine  of  its  comple- 
ment ;  the  cosine  of  an  angle  equals  the  sine  of  its  complement ; 
with  similar  relations  for  the  other  functions. 

Let  the  student  state  them. 

All  these  relations  are  comprised  in  the  general  statement : 
Any  function  of  an  acute  angle  equals  the  cofunction  of  its 
complement. 

5.  Relations  between  the  Functions.  Formulas.  From  the 
definitions  in  Sec.  4  we  see  that  sin  A  and  esc  A  are  reciprocals 
of  each  other,  or 

sin  A  = ,  esc  A  =  -: — - .  (1) 

esc  A  sin  A  7 

Also,  cos  A  and  sec  A  are  reciprocals,  or 

cos  A  = ,  sec  A  = T  •  (2) 

sec  A  cos  A  w 

Also,  tan  A  = ,  cot  A  = .  .  (3) 

'  cot  A  tan  A  v  ' 


■trvt 


fa  *  *  *  j^ ^ 


TRIGONOMETRIC    FUNCTIONS  6 


EXERCISES 

1.  If  the  sine  of  an  angle  is  f ,  what  is  its  cosecant  ? 

2.  If  tan  A  =  $ ,  what  is  the  value  of  cot  4? 

3.  Given  sec  A  =  %,  required  cos  A. 

6.    From  the  definitions  in  Sec.  4  we  have,  also, 


sin  A       c 
cos  A       b 

-  =  tan  A 
b 

c 

.'.  tan  A  = 

sin  A 
cos  A 

b 

COS  A         C 

sin  A       a 

-  =  cot  A. 

a 

c 

.'.  cot  A  = 

cos  A 

(4) 


<6> 

Formula  (4)  may  be  read :  the  sine  of  an  angle  divided  by 
its  cosine  equals  its  tangent ;  or,  briefly,  sine  divided  by  cosine 
equals  tangent. 

Express  formula  (5)  in  a  similar  way. 

7.    From  the  right   triangle  ACB   (Fig.   2)    we   have,  by 

geometry, 

a2  -f  b2  =  c2. 

Dividing  each  member  of  this  equation  by  c2,  we  have 

2 


(f)H')' 


1. 


or  (sin  A)2  +  (cos  A)2  =  1, 

or,  as  it  is  usually  written, 

sin2  A  +  cos2  A  =  1.  (6) 


6  PLANE    TRIGONOMETRY 

Again,  dividing  each  member  of  the  above  equation  by  b2, 
we  have 


(!)'"'-(!)' 


or  1  +  tan2  A  =  sec2  A.  (7) 

In  the  same  equation  divide  by  a2  and  prove,  also, 

l  +  cot2A  =  csc2A.  (8) 

As  we  have  seen,  sin  A  is  a  number;  sin2  A  means  the 
square  of  that  number. 

Formula  (6)  may  be  read :  the  square  of  the  sine  of  an 
angle  plus  the  square  of  its  cosine  equals  1;  or,  briefly,  sine 
square  plus  cosine  square  equals  1. 

Express  formulas  (7)  and  (8)  in  a  similar  way. 
Formulas  (1)  to  (8)  should  be  memorized. 

EXERCISES 

1.  Given  cos  A  =  |§,  find  sin  A  by  formula  (6). 

2.  If  sin  A  =  f ,  find  tan  A  by  formulas  (6)  and  (4) ;  use 
your  result  to  find  sec  A  by  formula  (7). 

3.  If  sec  A  —  -y~,  find  tan  A. 

4.  Write  down  formulas  (1)  to  (8)  from  memory  and  give 
the  proof  for  each. 

8.    Functions  of  45°,  60°,  30°.     The  numerical  values  of  the 
B    functions  of  45°,  60°,  30°  can  readily  be 
found. 

To  find  the  functions  of  45°,  construct 
a  right  angle,  C,  and,  adopting  any  con- 
venient unit,  set  off  CA  =  CB  =  1. 
Then,      A  =  45°,  AB  =  V2.     why? 
Therefore, 

sin  45°  =  — 7=  >  cos  45°  =  —7=  ? 

V2  V2 


TRIGONOMETRIC    FUNCTIONS 


tan  45°  =  j  =  l, 
cot  45°  =  1, 


V2         r~ 
Sec45°  =  — =V2, 

csc45°=V2. 


To  find  the  functions  of  60°  and  30°,  construct  an  equi- 
lateral* triangle,  ABC  (Fig.  4),  hav- 
ing each  side  equal  to  2.  Then 
each  of  its  angles  equals  60°.  Draw 
BD  perpendicular  to  A  C.  By  geom- 
etry AD  =  1,  and  BD2  =  AB*  -  AD2, 
from  which  BD  =  V3.  Then,  from 
the  right  triangle  ADB,  since 

Z  A  =  60°, 
we  have 

V3 


sin  60°  = 


cos  60c 


esc  60°  = 


tan  60°  =  V3, 
Again,  BD  bisects  the  angle  ABC  [why?],  giving 
Z  ABB  =  30°, 


and  we 

have 

sin  30°  =  ^ 

cot  30°  =  V3, 

V3 
cos  30°  =  -^, 
z 

2 
sec  30°  =  — F 
V3 

tan  30°  =  — = » 

esc  30°  =  2. 

V3 

The  student  should  familiarize  himself  with  the  right  triangles  in  Figs.  3 
and  4  and  thus  be  able  readily  to  recall  the  functions  of  the  angles  considered. 


PLANE   TRIGONOMETRY 


EXERCISES 


1.  Deduce  the  values  of  the  functions  of  45°  by  using  a 
right  triangle  having  each  leg  equal  to  a. 

2.  Construct  an  equilateral  triangle  having  each  side  equal 
to  a,  draw  the  perpendicular  as  in  Fig.  4,  and  find  the  func- 
tions of  60°  and  30°. 

3.  Verify  formulas  (1)  to  (8)  for  the  angles  45°,  60°,  30°. 
Thus,  making  A  =  60°,   (6)   becomes  sin2  60°  +  cos2  60°  =  1. 

Substituting  the  values  found  in  this  section, 


^(0'=i,  <*  W- 


Fig.  5 


\  2  J       \2J        '   -       4      4 

which  verifies  formula  (6)  for  the  angle  60°. 

9.    If  one  of  the  functions  of  an  acute  angle  be  given,  the 
angle  can  be  constructed,  and  the  remain- 
ing functions  can  be  found.     For  example, 
let  it  be  required  to  construct  the  acute 
angle  which  has  f  for  its  sine.     Construct 
the  right  triangle  ACB  (Fig.  5),  having 
BC  =  3,  AB  =  5. 
How  is  this  done  ? 
Then  Z  A  is  the  required  angle,  for  sin  A  —  f . 
To  find  the  remaining  functions,  we  have  AC  =  25  —  9,  or 
AC  =  4.     Therefore, 

cos  A  =  £ ,    tan  A  =  j,    cot  A  =  |,    sec  A  =  |,    esc  A  =  ^. 
The  same  results  could  have  been  obtained  by  using  for  BC 
and  AB  any  two  numbers  proportional  to  3  and  5. 
Try  6  and  10. 

EXERCISES 

Construct  the  angle  A  and  find  the  remaining  functions, 
having  given : 
1.  cos  A  =  T%.     2.  esc  A  =  -1/-.     3.  tan  A  =  —    4.  cot  A  =  T5^. 


TRIGONOMETRIC   FUNCTIONS  9 

» 

MISCELLANEOUS  EXERCISES 

m     _>  sec  x 

1.  Prove  =  tan  x. 

cscx 

„     «  1  +  tan  x        . 

2.  Prove  =  sm  x  -f  cos  x. 

secx 

3.  Prove  (sin  x  +  cos  x)2  +  (sin  x  —  cos  x)2  =  2. 

4.  Which  is  greater,  sin  42°  or  cos  42°  ?   tan  54°  or  cot  54°  ? 

5.  Given  sec  A  =  § ,  find  the  remaining  functions  by  formulas  (1) 
to  (8). 

6.  Given  tan^i  =  T8-,  find  the  remaining  functions  by  formulas  (1) 
to  (8). 

7.  Given  tan  A  =  t  =  - ,  find  the  remaining  functions. 

8.  Each  side  of  an  equilateral  triangle  is  14.     Find  the  altitude  by 
using  one  of  the  functions  of  60°. 

9.  The  hypotenuse  of  a  right  triangle  is  26  and  one  angle  is  30°.    Find 
the  length  of  each  leg,  correct  to  four  places  of  decimals. 

10.  In  a  right  triangle  one  angle  is  60°  and  the  opposite  side  is  8.  Find 
the  other  leg  and  the  hypotenuse. 

11.  A  ladder  stands  against  a  wall  and  makes  an  angle  of  60°  with 
the  ground.  Find  the  length  of  the  ladder  if  it  reaches  a  window  48  feet 
from  the  ground. 

12 .  Using  the  functions  of  30°,  45°,  and  60°,  find  the  values  of  the  follow- 
ing expressions  :     (a)  g  gin  3()0  _  g  cog  6QO  +  tan  ^0 

(b)  8  sin  60°  +  12  cos  30°  -  2  tan  45°. 

(c)  5  sin  30°  -  7  cos  60°  +  3  tan  45°. 

(d)  5  cos  30°  -  8  cos  60°  -  f  VI. 

13.  When  cos  A  =  ^,  find  the  value  of  3  sin  A  —  4  sin8  .A. 

14.  When  cos  A  =  I,  find  the  value  of 

*'  cot  A 

2  tan  A 


15.  When  cos  A  =  A  V§,  find  the  value  of 

11  1  -  tan2  A 

16.  Express  each  function  of  the  angled,  in  terms  of  sin  A. 

x2      v2 

17.  When  x  =  a  cos  6,  y  =  b  sin  0,  show  that  —  -f  —  =  1. 

a2      b2 


CHAPTER   II 
LOGARITHMS 

10.  If  ax  =  n}  the  exponent  x  is  called  the  logarithm  of  n 
to  the  base  a. 

So  in  the  equation  73  =  343,  the  exponent  3  is  the  logarithm 
of  343  to  the  base  7. 

Definition.  With  reference  to  any  base,  the  logarithm  of  a 
number  is  the  exponent  of  the  power  to  which  the  base  must  be 
raised  to  produce  the  given  number. 

To  ask,  What  is  the  logarithm  of  1296  to  the  base  6  ?  is  the 
same  as  to  ask,  To  what  power  must  the  base  6  be  raised  in 
order  to  produce  1296  ? 

The  equation  ax  =  n  is  said  to  be  in  the  exponential  form ; 
such  an  equation  can  always  be  written  in  the  equivalent 
logarithmic  form  \ogan  =  x,  which  is  read:  logarithm  ofn  to 
the  base  a  equals  x. 

Any  positive  number  except  1  may  be  used  as  the  base  of  a 
system  of  logarithms. 

EXERCISES 

1.  The  base  being  4,  what  are  the  logarithms  of  the  follow- 
ing numbers  ?     64,  1024,  4096,  2. 

2.  What  are  the  values  of  the  following  ?  log5  625,  log3  81, 
log749,log9729. 

3.  Express  the  following  equations  in  the  logarithmic  form : 
88  =  512,  25  =  32,  108  =  1000,  10~8  =  0.001,  101  =  10,  10°  =  1, 

103.52543  =  3353^   cy  _  ^   #~V  =  ^ 

4.  Express  the  following  in  the  exponential  form  : 

log6  216  =  3,  log9  81  =  2,  log,  m  =  y, 

\ogaPq  =  y  +  z,       log1010000  =  4,      log10326.=  2.51322. 

10 


LOGARITHMS  11 

11.  Laws  of  Exponents.  Properties  of  Logarithms.  Since 
logarithms  are  exponents,  it  may  be  well  to  recall  the  laws  of 
exponents  as  given  in  algebra. 

We  have,  for  all  values  of  a,  x,  y,  etc., 

ax>ay  =  ax  +  tf,  thus  az-a^  —  a}\ 

ax  +  a"  =  ax~v,  "  a8  -=-  az  =  a5 ; 

(axy  =  a*x,  "  (a*y  =  a12; 

a*  =a  Va,  "  ai  =  Va  ; 

a9  =  vV  =  (  Valy,     thus     ai  =  V^  =  (  V^)8 ; 

a-*  =  —?  thus  a-5  =  -=  > 

a8  a6 

and,  for  any  value  of  a  except  0, 

a0  =  1,  thus  13°  =  1. 

From  these  laws  are  derived  the  following  properties  of 
logarithms : 

(a)  The  logarithm  of  the  product  of  two  numbers  equals  the 
logarithm  of  the  first  plus  the  logarithm  of  the  second. 

Tor  example,  log  33  =  log  3  +  log  11. 

Proof.    Let  a  be  the  base  and  m  and  n  the  two  numbers,  and 

let 

logara  =  x, 

and  logarc  =  y. 

Then,  by  Sec.  10,  ax  =  m. 

By  multiplication,  a*+,  =  mn' 

Therefore,  by  Sec.  10,  logara??  =  x  -f  y. 
For  x  and  y  substitute  their  values,  and  we  have 
logara?i  bb  logara  +  logan. 


12  PLANE   TRIGONOMETRY 

Corollary.     The  logarithm  of  the  continued  product  of  any 
number  of  factors  equals  the  sum  of  their  logarithms. 
Thus,  let  m  =  cd. 

Then,  log  cdn  =  log  mn  =  log  m  +  log  n  =  log  (cd)  +  log  n 
=  log  c  +  log  d  -f  log  n. 

(b)   The  logarithm  of  a  fraction  equals  the  logarithm  of  the 
numerator  minus  the  logarithm  of  the  denominator. 
Thus,  log  -V-  =  log  17  —  log  5. 

7YV 

Proof.     Denote  the  fraction  by  —  and  let 

J    n 


\ogam  =  x, 

and                                         logan  =  y. 

Then,                                     ax  =  m, 

and                                               ay  =  n. 

By  division,                       ax~v  =  —  • 
J                 '                                     n 

Therefore,  by  Sec.  10,  logtt  —  =  x  —  y, 

or                                          loga^  =  log0m- 

"  loga^ 

(c)    The  logarithm  of  a  power  of  a  number  equals  the  loga- 
rithm of  the  number  multiplied  by  the  exponent  of  the  power. 
Thus,  log  2S5  =  5  log  23. 

Proof.     Let  n  be  the  number  and  p  the  exponent  of  the 

power,  and  let 

loga?i  =  x. 

Then,  ax  =  n. 

Raising  each  member  of  this  equation  to  the  power  p, 

apx  ==  np. 

Therefore,  loganp=px, 

or,  substituting  for  x  its  value, 

\oganp  =  p  \ogan. 


LOGARITHMS 


13 


(d)   The  logarithm  of  a  root  of  a  number  equals  the  loga- 
rithm of  the  number  divided  by  the  index  of  the  root. 


Thus, 


log^3963  =  l2£f63. 


Proof.     Let  n  be  the  number  and  r  the  index  of  the  root. 
Then,  log0-v^  =  log0?z'  =  -  logarc  =  -^f^  ■ 

(e)  Since  a1  =  a,  the  logarithm  of  the  base  is  unity. 

(f)  Since  a0  =  1,  the  logarithm  of  1  is  0. 

(g)  If   a>l,a-a=-  =  -  =  0.     ■.-.Iog0=-oo    in   any 
system  in  which  the  base  is  greater  than  1. 


EXERCISES 


0.84510,  find 


Given  log  2  =  0.30103,  log  3  =  0.47712,  log  7 
the  logarithms  of  the  following  numbers : 

6,  18,  32,  192,  if  2,  *#£-,  76,  365,   Vl26,   V^86,   ^486. 

12.  Common  System  of  Logarithms.  The  common  system  of 
logarithms  is  that  in  which  the  base  is  10.  When  no  base  is 
indicated,  it  is  understood  to  be  10.  It  is  easy  to  find  the 
common  logarithms  of  numbers  which  are  integral  powers 
of  10. 


Thus, 

10°  =  1. 

•.logl           =0.. 

101  =  10. 

•.log  10         =1. 

102  =  100. 

'.log  100       =2. 

103  =  1000. 

\  log  1000     =  3. 

104  =  10000. 

•.log  10000  =4. 

Again, 

io-1  -  A      «  o.i. 

•.log  0.1        =-1. 

io-2  =  Tho     =  o.oi. 

•.log  0.01      =-2. 

io-3  =  T*v*  =0.001. 

'.log  0.001    =-3. 

io-4  =  t^o*  =  o.oooi. 

•.log  0.0001  =  -4. 

/ 


14  PLANE   TRIGONOMETRY 

Consider  next  the  number  257.  It  lies  between  100  and 
1000.  The  logarithm  of  257,  therefore,  lies  between  2  and  3. 
By  methods  explained  in  algebra  it  is  found  to  be,  approx- 
imately, 2.40993.  Similarly,  the  logarithm  of  any  positive 
number  consists  of  a  whole  number  plus  a  decimal.  The 
whole  number  is  called  the  characteristic,  and  the  decimal 
part  is  called  the  mantissa. 

Table  I  in  the  Appendix  gives  the  mantissas  of  the  loga- 
rithms of  numbers  from  1  to  10000.  It  is  not  necessary  to 
give  the  characteristics  in  the  table,  for  they  may  be  found  by 
inspection,  as  follows :  Consider  any  positive  number  that 
has,  say,  three  digits  to  the  left  of  the  decimal  point, 
457.62  for  example.  It  lies  between  100  and  1000 ;  its  loga- 
rithm, therefore,  is  greater  than  2  and  less  than  3.  Hence, 
log  457.62  =  2  -f-  a  decimal,  and  the  characteristic  is  2. 

So  any  number  that  has  four  digits  to  the  left  of  the  deci- 
mal point,  as  5687.037,  lies  between  1000  and  10,000;  its 
logarithm  is  therefore  greater  than  3  and  less  than  4.  Hence, 
log  5687.037  =  3  -}-  a  decimal,  and  the  characteristic  is  3. 

For  numbers  greater  than  1,  therefore,  we  see  that  the  char- 
acteristic is  positive  and  1  less  than  the  number  of  digits  to 
the  left  of  the  decimal  point.  ■ 

Any  positive  number  less  than  1  may  be  expressed,  exactly 
or  approximately,  as  a  decimal,  and  the  characteristic  of  its 
logarithm  determined  as  follows  :  Consider  the  decimal  0.0068. 
It  is  greater  than  0.001  and  less  than  0.01.  The  logarithm  of 
0.001  is  —  3,  and  the  logarithm  of  0.01  is  —  2.  Hence,  the 
logarithm  of  0.0068  is  greater  than  —  3  and  less  than  —  2 ;  i.e., 

log  0.0068  =  —  3  plus  a  decimal, 
or  log  0.0068  =  —  2  minus  a  decimal.- 

It  will  be  found  convenient  to  adopt  the  first  form,  keeping 
the  decimal  part  of  the  logarithm  always  positive. 
Hence,  the  characteristic  of  log  0.0068  is  —  3. 


LOGARITHMS  15 

Similarly,  0.0007  is  greater  than  0.0001  and  less  than  0.001 ; 

and  since 

log  0.0001  =  -  4, 

and  log  0.001  =  -  3, 

it  follows  that  log  0.0007  is  greater  than  —  4  and  less  than 

—  3.     Hence, 

log  0.0007  =  —  4  pins  a  decimal ; 

the  characteristic  is  therefore  —  4. 

We  see,  then,  that  if  a  number  is  entirely  decimal,  the 
characteristic  is  negative,  and  numerically  1  greater  than  the 
number  of  ciphers  immediately  following  the  decimal  point. 

EXERCISE 

Determine  the  characteristic  of  the  logarithm  of  each  of 
the  following  numbers  :  368,  42,  4687,  5238.64,  23.1,  13.42, 
6.342,  0.0063,  0.0008,  0.81,  0.43. 

Next,  suppose  that  the  mantissa  of  the  logarithm  of  2681  is 
given  to  be  .42830 ;   by  the  rule  the  characteristic  is  3. 
.'.log  2681  =  3.42830. 
Suppose  the  mantissa  of  the  logarithm  of  0.0634  is  given  to 
be  .80209 ;  by  the  rule  the  characteristic  is  —  2.     Then  we 
have,  keeping  the  mantissa  positive, 

log  0.0634  =-2  +  .80209, 
or,  more  compactly, 

log  0.0634  =  2.80209, 
where  the  minus  sign  applies  only  to  the  characteristic. 

Instead  of  the  negative  characteristics,  —  1,  —  2,  —  3,  etc., 
it  is  often  more  convenient  to  write  their  equals,  9  —  10, 
8  -  10,  7  -  10,  etc.     Thus, 

log  0.634  =  1.80209  =  9.80209  -  10, 
log  0.0634  =  2.80209  =  8.80209  -  10, 
log  0.00634  =  3.80209  =  7.80209  -  10. 


16  PLANE   TRIGONOMETRY 

By  inspection  of  these  results  it  is  seen  that  if  a  number  is 
entirely  decimal,  the  characteristic  of  its  logarithm  may  be 
found  by  subtracting  the  number  of  ciphers  immediately  fol- 
lowing the  decimal  point  from  9  and  annexing  —  10  to  the 
result. 

13.  Logarithms  of  Numbers  having  the  Same  Sequence  of  Digits. 
If  two  numbers  have  the  same  sequence  of  digits,  i.e.,  if  they 
differ  only  in  the  position  of  the  decimal  point,  their  logarithms 
have  the  same  mantissas. 

Thus,  log  26.834  and  log  2683.4  have  the  same  mantissa ;  for 
2683.4  =  26.834  x  100. 

Therefore,  by  Sec.  11,  (a), 

log  2683.4  =  log  26.834  +  log  100  =  log  26.834  +  2 ; 
and  adding  the  whole  number  2  to  the  logarithm  of  26.834 
does  not  affect  its  mantissa. 

In  general,  if  two  numbers  differ  only  in  the  position  of  the 
decimal  point,  the  greater  is  equal  to  the  less  multiplied  by 
10,  100,  1000,  or  some  other  integral  power  of  10.  The  loga- 
rithm of  the  greater,  therefore,  equals  the  logarithm  of  the 
less  increased  by  1,  2,  3,  or  some  other  whole  number,  and 
the  addition  of  a  whole  number  to  the  logarithm  of  the  less 
does  not  affect  its  mantissa. 

The  statement  that  the  mantissas  are  the  same  for  all 
numbers  differing  only  in  the  position  of  the  decimal  point 
is  true  only  when  the  decimal  part  of  every  logarithm  is  kept 
positive.  For,  if  we  consider  the  number  .26834,  which  has 
the  same  sequence  of  figures  as  2683.4,  we  have 

log  .26834  =  1.42868  =  -  1  +  .42868  =  -  .57132. 

In  the  last  form  the  decimal  part  of  the  logarithm  has  not 
been  kept  positive,  and  does  not  agree  with  the  decimal  part  of 
log  2683.4.  The  operations  of  adding  and  subtracting  loga- 
rithms are  also  made  simpler  by  keeping  the  decimal  part 
positive. 


LOGARITHMS  17 


EXERCISES 

1.  Given  log  326.82  =  2.51431,  write  down  the  logarithms 
of  the  following  numbers:  32682,  32.682,  3.2682,  0.32682, 
3268.2,  0.00032682,  326820,  3268200. 

2.  Write  down  three  numbers,  each  greater  than  1,  and 
three  numbers,  each  less  than  1,  such  that  their  logarithms 
have  the  same  mantissas  as  the  logarithm  of  5468. 

3.  Make  the  decimal  part  positive  in  the  following  loga- 
rithms :  -  2.36823,  -  1.21656,  -  3.42857.     Thus, 

-  2.36823  =  -  2  -  .36823  =  -2-1  +  1-  .36823 
=  -  3  +  .63177  =  3.63177. 

4.  Make  the  decimal  part  negative  in  the  following  :  2.18734, 
1.21683,  3.46827.     Thus, 

2.18734  =  -  2  +  .18734  =  -  1.81266. 

5.  Determine  the  base  in  each  of  the  following  cases: 

loga8  =  1.25 ;  log6i  =  1.25  ;  loge  A  -  -  1-25. 
1  I  6 

MISCELLANEOUS  EXERCISES 

Read  the  explanation  of  Table  I  in  the  Appendix,  and  then  use  loga- 
rithms to  perform  the  operations  indicated  in  the  following  exercises. 

^     xv   *  *.        i        m  1368.34  x  267 

1.    Find  the  value  of 

835 

1368.34  x  267 


Solution.    Let 

835 

Then,  by  Sec.  11,  (a)  and  (b), 

log  x  =  log  1368.34  +  log  267  -  log  835. 

log  1368.34  =  3.13620 

log  267  =  2.42651 1 

5.56271 

log  835  =  2.92169 

log  x  =  2.64102 

3  =  437.54.     , 


18  PLANE    TRIGONOMETRY 


«    ^.   ***.        i        x  1760x^86834 

2.   Find  the  value  of 


19.13  x  27* 


„'      ,  T  1760  x  V86834 

Solution.     Let  x  = 5 — 

19.13  x27* 

By  Sec.  11,  logo;  =  log  1760  +  log^834  _  (log  19.13  +  flog  27). 

log  1760  =  3.24551 

tV  log  86834  =  0.32925 

3.57476 

2.35523 

logx  =  1.21953 

x  =  16.578. 


log  19.13  =  1.28171 

flog  27  =  1.07352 

2.35523 


3.  Find  the  value  of  x  from  the  equation  x  =  ^VsV 
Solution.     We  have        log  x  =  log  183  -  log  3687. 

The  subtrahend  in  this  case  being  greater  than  the  minuend,  we  add 
10  to  the  logarithm  of  183  and  subtract  10  from  it,  in  order  to  avoid  a 
negative  mantissa,  writing 

log  183  =  12.26245  -  10 
log  3687  =    3.56667 

logx=    8.69578-10 
x  =  0.04963. 

4.  Find  the  value  of  the  7th  root  of  0.08683. 


Solution.     Let  x  =  Vo.  08683. 

log  0.08683 
Then,  log  x  =     " 

log  0.08683=    8.93867  -  10 

60  60 

7)68.93867-70 

logx=    9.84838-10 

x  =  0.7053. 

Here  we  add  and  subtract  60  in  order  that  the  quotient  obtained  by 
dividing  the  negative  part  of  log  0.08683  by  7  may  be  10. 


% 


m«v 


LOGARITHMS 


19 


5. 


6. 


263  x  178 . 
341 

37816 
231  x  17  ' 

0.82134  x  276 


0.69473 
0.01368  x  (1478)* 


7. 
8. 

9.  Vo.  06834 

10.  ^168342. 

11 


256.8  x  (19)*  x  V1283 
1863  x  23.12 


12. 
13. 

14. 
15. 
16. 
17. 
18. 

19. 


12.359  x  0.17954 
0.00238 

(8.17)3  x  (0.236)2  x  536 

^129347 

26843.9  x  0.0063 
18.7  x  68.492 

186  x  27924  x  0.00123. 

12963 
1.3  x  1.8  x2.l' 

237 


46.1  x  963 
•v/0.0063  x  (52.61)1 


Ans.  137.29. 

Ans.  9.6298. 

Ans.  326.29. 

Ans.  777.32. 

Ans.  0.5847. 

Ans.  2.0296. 

Ans.  28.891. 
Ans.  932.33. 
Ans.  1546.3. 

Ans.  0.13204. 
Ans.  6388.6. 
Ans.  2638.1. 
Ans.  0.014109. 
Ans.  0.0053385. 

Ans.  358.15. 


CHAPTER   III 


SOLUTION   OF   RIGHT   TRIANGLES 

14.    A  right  triangle  can  be  solved  if,  besides  the  right  angle, 
any  two  parts  be  given,  provided  one  of  them  be  a  side. 
Let  the  right  triangle  be  ACS  (Fig.  6).     We  have 


sin  A  = 


a 


cos  A  = 


tan  A  = 


Since  A  +  B  =  90°,  if  either  A  or  B  be  one  of  the  given 
parts,  the  other  acute  angle  is  at  once 
B  known. 

We  may,  therefore,  in  any  case  select 
from  the  three  equations  above  one  which 
contains  only  one  unknown  quantity  and 
solve  for  that  unknown.  Then  select 
another  which  contains  only  one  unknown  quantity  and  solve 
as  before. 

Before  proceeding  farther  the  student  should  read  the  explanation  of  the 
Tables  of  Trigonometric  Functions  in  the  Appendix. 


EXAMPLES 
1.    Given  A  =  38°,  c  =  187,  to  find  B,  a,  b. 


Solution. 


sin  A 


B  =  90°  -  A  =  52°. 
a 


or 


bb  sin  38°. 

187 

.-.  a  =  187  sin  38°. 

.-.  log  a  =  log  187  +  log  sin  38°. 

20 


SOLUTION   OF   RIGHT   TRIANGLES  21 

log  187  =  2.27184 
log  sin  38°  =  9.78934  -  10 
log  a  =  2. 061 18 
a  =115.127. 

We  have  also  —  =  cos  38°, 

187 

or  b  =  187  cos  38°. 

log  b  =  log  187  +  log  cos  38°. 
log  187  =  2.27184 
log  cos  38°  =  9.89653  -  10 
log  6  =  2.16837 
b  =  147.36. 
The  accuracy  of  the  work  should  be  tested  by  solving  by  other  for- 
mulas, or  by  seeing  whether  the  parts  of  the  triangle  as  found  satisfy  other 
relations  which  must  subsist  if  the  solution  is  correct.      Any  formula 
involving  one  or  more  of  the  required  parts  not  used  in  the  solution  may 
be  used  as  a  check  formula.     Thus,  having  used  in  the  solution  two  of 
the  three  equations  above,  the  third  may  be  used  as  a  check  formula. 

Then  a  and  b  ought  to  satisfy  the  relation  tan  A  =  - ,  from  which 

b 
log  tan  A  =  log  a  —  log  b. 

log  a  =  12.06118  -  10 
log  b  =    2.16837 
.-.log  tan  A  =    9.89281  -  10 
and  from  the  table  we  find 

log  tan  A  =  log  tan  38°  =  9.89281  -  10. 
The  two  results  agree. 
Another  check  formula  is  a?  =  c2  —  62, 
or  a?  =  (c  +  b)  (c  -  b). 

Applying  logarithms,  2  log  a  =  log  (c  -f  6)  -f  log  (c  —  b) 
=  log  334.357  +  log  39.643. 
log  334. 357  =  2.51421 
log  39.643  =  1.59816 
2  log  a  =  4.12237 
log  a  =  2.06118. 
This  agrees  with  the  value  of  log  a  as  determined  above.     In  some 
cases  there  may  be  a  slight  discrepancy,  due  to  the  fact  that  the  values 
given  in  the  tables  are  only  approximately  correct ;  correct,  that  is,  only 
as  far  as  five  places  of  decimals. 


PLANE   TRIGONOMETRY 


j  2.    Given  A  =  67°,  b  =  148,  to  find  B,  a,  c. 

B  =  90°  -  A  -  23°, 

-  =  tan  A, 
b 

or  —  =  tan  67°. 

148 

.-.  log  a  =  log  148  +  log  tan  67°. 

log  148  =  2.17026 

log  tan  67°  =  0.37215 

loga  =  2.54241 

a  =  348.67. 

To  find  c,  cos  A  =  - , 

c 

-*«      148 

or  cos  67°  = 

c 

log  cos  67°  =  log  148  —  log  c. 

log  c  =  log  148  —  log  cos  67°. 
log  148  =  12.17026 -10      . 
log  cos  67°  =    9.59188  -  10 
logc=    2.57838 
c  =  378.77. 
Checks,   sin  A  =  - ,  cfi  =  (c  +  b)(c  —  b). 
Apply  these  tests. 

\)3.    Given  a  =  327,  c  =  578,  to  find  A,  B,  b. 

sin  A  =  - , 
c 


or 


log  sin  A  =  log  327  —  log  678. 
log  327  =  12.51455-  10 
log  578  =    2.76193 
log  sin  A  =    9.75262  -  10 
A  =  34°  27'  13". 
B  =  90°  -  A  =  55°  32'  47' 


To  find  6,  we  have  -  =s  cos  A. 


c 
log  b  =  log  c  +  log  cos  A. 


SOLUTION   OF   RIGHT   TRIANGLES  23 

log  578  =  2.76193 
log  cos  34°  27'  13"  =  9.91623  -  10 
log  6  =  2.67816 
b  =  476.61. 

Checks,    tan  A  =  -  ,  o2  =  (c  +  a)  (c  -  a),  or  &2  =  905  x  251. 
o 

.-.  2  log  6  =  log  905  +  log  251. 
log  905  =  2.95665 
log  251  =  2.39967 
2  log  6  =  5.35632 
or  log  b  =  2.67816,  as  above. 

Apply  the  other  test. 

4.    Given  a  =  26,  b  =  37,  to  find  ^,  B,  c. 

™    ,  a      26 

We  have  tan  ^4  =  -  =  — . 

b      37 

.-.  log  tan  A  =  log  26  —  log  37. 

log  26  =  11.41497  -  10 

log  37  =    1.56820 

log  tan  A=    9.84677-10 

A  =  35°  5'  44". 

B  =  90°  -  A  =  54°  54'  16". 

To  find  c,  we  have  -  =  sin  A. 

.-.  log  a  —  log  c  =  log  sin  A, 
or  log  c  =  log  a  —  log  sin  A. 

log  26  =  11.41497  -  10 
log  sin  35°  5'  44"  =    9.75962-10 
logc=    1.65535 
.-.  c  =  45.22. 

Check,   cos  A  =  -  • 
c 

The  student  will  find  it  convenient  to  remember  the  following  relations 
obtained  from  the  formulas  at  the  beginning  of  this  section: 

a=c  sin  A,  b  =  ccosA,  a=b  tan  A. 

15.    The  solution  may  be  effected  in  any  case,  also,  without 
the  use  of  logarithms. 


24  PLANE   TRIGONOMETRY  ^ 


EXAMPLE 
X.  Given  A  =  32°,  c  =  200,  to  find  a,  b,  B. 

We  have  -  =  sin^4, 

c 

or  —  =  sin  32°. 

200 

.-.  a  =  200  x  sin  32°. 

From  the  Table  of  Sines,  we  find 

sin  32°  =  0.5299. 

.-.  a  =  0.5299  x  200, 

or  a  =  105.98. 

Also,  b  =  200  x  cos  32°  =  200  x  0.8480, 

or  b  =  169.60. 

B  =  90°  -  A  =  58°. 

EXERCISES 

Solve  the  following  right  triangles,  checking  the  results 
when  the  answers  are  not  given : 

1.    Given  a  =  43,  c  =  58,  to  find  A,  B,  b. 

Ans.  A  =  47°  50'  55",  B  =  42°  9'  5",  b  =  38.924. 

^2.    Given  a  =  31,  b  =  98,  to  find  A,  B,  c. 

Ans.  A  =  17°  33'  12",  B  =  72°  26'  48",  c  =  102.79. 

3.  Given  A  =  30°  12',  c  =  288.6. 

Ans.  a  =  145.173,  b  =  249.43. 

4.  Given  a  =  146.83,  c  =  197.14. 

Ans.  A  =  48°  8'  33",  b  =  131.55. 

5.  Given  a  =  0.1442,  b  =  0.1647. 

Ans.  A  =  41°  12'  14",  c  =  0.21891. 

6.  Given  A  =  68°  23',  a  =  223.95. 

Ans.  b  =  88.744,  c  =  240.89. 

7.  Given  &  =  168.92,  c  =  289.64. 

Ans.  B  =  35°  40'  33",  a  =  235.28. 


SOLUTION   OF   RIGHT   TRIANGLES 


25 


8.    Given  B  =  64°  38 ',  b  =  38.21. 

Ans.  c  =  42.287,  a  =  18.116. 
U  9.    Given  a  =  42,  c  =  145. 


10. 


Fig.  7 


^n*.  ^  =  16°  50'  14",  b  =  138.78. 
Given  b  « 12,  c  =  13. 

-4w«.  £  =  67°  22'  48",  a  =  5. 
Given   J  =  35,  c  =  37. 
Given  4  =  63°  12',  c  =  128. 
Given  B  =  47°  17',  a  =  23.48. 
Given  A  =  27°  36',  b  =  468. 
Given  a  =  123,  b  =  178. 

16.  Definitions.  In  Fig.  7,  let  A  and  5  be  any  two  points  of 
which  B  is  at  the  greater  height  above 
the  sea  level.  In  the  vertical  plane 
through  A  and  B  draw  the  horizontal 
lines  A  C,  BD.  Then,  the  angle  CAB  is 
called  the  angle  of  elevation  of  B  with 
reference  to  A ;  and  the  angle  DBA  is 
called  the  angle  of  depression  of  A  with  reference  to  B. 

The  angle  of  elevation  of  a  heavenly  body  is  called  its 
altitude. 

The  acute  angle  which  a  line  on  the  earth's 
B^  B       surface  makes  with  a  meridian  (i.e.,  a  north- 

^^  /  and-south  line)  drawn  through  its  initial 
point  determines  the  bearing  of  the  line. 
Thus,  in  Fig.  8,  NS  is  a  meridian ;  the  bear- 
ing of  AB  is  K  30°  E.,  which  is  read  north 
30°  east ;  the  bearing  of  CD  is  S.  46°  E. ;  of 
^^*         3   MP,  S.  77°  W. ;  of  QR,  N.  60°  W. 

Heights   and   Distances.      Many   practical 

problems  in  which  heights    and   distances 

S  are  to  be  found  may  be  solved  by  the  use 

Fro.  s  of  right  triangles  if  we  regard  the  earth's 


N 


26  PLANE   TRIGONOMETRY 

surface  as  a  plane.     Examples  will  be  found  at  the  end  of 
0  this  chapter. 

17.  Area  of  a  Regular  Polygon.  Let  AB 
(Fig.  9)  be  a  side  of  a  regular  polygon 
of  n  sides ;   O  the  center  and  r  (=  OC) 


the  radius  of  the  inscribed  circle,  S  the 
area  of  the  polygon ;    to  find  the  value 
of  S  in  terms  of  n  and  r. 

,  *~        360°  ,  in„      180° 

Now,  Z.AOB  = .'.ZAOC 


n 
180c 


Also,  area  of  A  A  OB  =  OC  X  AC  =  r  X  r  tan 

and  as  there  are  n  such  triangles,  we  have 

180° 
S  =  nr2  tan 


EXERCISES 

1.  Express  S  in  terms  of  a  and  n,  where  a  =  AB. 

2.  Find  the  perimeter  in  terms  of  r  and  n. 

Find   in   terms  of  r  the   areas   of   the   following   regular 
polygons : 

3.  Triangle.  5.    Hexagon.  7.    Dodecagon. 

4.  Square.  6.    Octagon. 


MISCELLANEOUS  EXERCISES 

1.  The  bounding  lines  of  a  tennis  court  include  a  rectangle  78  feet  by 
36  feet.     Find  the  angle  made  by  a  diagonal  and  the  shorter  side. 

Arts.  65°  13' 29". 

2.  Find  the  length  of  the  shadow  cast  by  a  vertical  rod  12  feet  long 
when  the  altitude  of  the  sun  is  78°  42'.  Am.  2.397  feet. 

3.  What  is  the  bearing  of  the  line  along  which  a  man  must  go  in  order 
to  reach  a  point  4  miles  east  and  9  miles  north  of  his  starting  point  ? 

Ans.  N.  23°  57'  46"  E. 


SOLUTION   OF   RIGHT   TRIANGLES  27 

4.  What  is  the  radius  of  a  circle  in  which  a  chord  100  feet  long  sub- 
tends an  angle  of  1°  at  the  center  ?  Ans.  5729.71  feet. 

5.  What  is  the  radius  of  a  circle  when  tangents  A  I,  BI,  300  feet  long, 
form  an  angle  AIB  =  126°  ?  Ans.  588.78  feet. 

Solve  without  logarithms. 

6.  The  radius  of  a  circle  is  2864.93  feet.  A  chord,  AB,  and  a  tangent, 
AC,  are  each  100  feet.     Find  the  length  of  BC.  Ans.    1.7453  feet. 

— s  7.    The  radius  of  a  circle  is  24.96  feet.     Find  the  perimeter  of  the  reg- 
ular inscribed  pentagon.  Ans.  146.71  feet. 

8.  What  is  the  angle  of  elevation  of  the  top  of  a  tower  146  feet  high 
from  a  point  325  feet  from  its  foot  ?  Ans.  24°  11'  20". 

Solve  without  logarithms. 

9.  A  train  started  at  10 :  15  a.m.  and  proceeded  on  a  straight  track  at 
a  uniform  rate  of  34  miles  per  hour.  At  11  :  45  a.m.  it  had  reached  a 
point  26  miles  further  west  than  the  starting  point.  Find  the  bearing  of 
the  track.  Ans.  N.  30°  39'  3"  W.,  or  S.  30°  39'  3"  W. 

10.  From  a  point  80  feet  from  the  base  of  a  tower  the  angle  of  eleva- 
tion of  an  object  in  one  of  the  windows  is  50°,  and  the  angle  of  elevation 
of  the  top  of  the  tower  is  58°  10'.  Find  the  distance  from  the  object  to 
the  top  of  the  tower.  •  Ans.  33.512  feet. 

Solve  without  logarithms. 

11.  The  radius  of  a  circle  is  50  feet.  Find  the  area  of  the  regular 
inscribed  pentagon.  Ans.  5944. 1  square  feet. 

12.  A  regular  octagon  is  inscribed  in  a  circle  whose  radius  is  49.92  feet. 
Find  the  length  of  the  side  and  of  the  apothem. 

Ans.  Side,  38.21  feet;  apothem,  46.12  feet. 

13.  Find  the  area  of  the  octagon  in  Example  12. 

Ans.  7048.33  square  feet. 

14.  A  chord  16  inches  long  subtends  at  the  center  of  a  circle  an  angle 
of  46°  24'.     Find  the  side  of  the  inscribed  square.     Ans.  28. 72  inches. 

15.  At  a  point  50  feet  from  the  base  of  a  tower  the  angle  of  elevation 
of  the  top  was  found,  by  the  use  of  a  transit,  to  be  68°  18'.  If  the  height 
of  the  instrument  was  4.75  feet,  find  the  height  of  the  tower. 

Ans.  130.39  feet. 


/ 


28  PLANE   TRIGONOMETRY 

*    16.    The  radius  of  a  circle  is  1910.1  feet.     Tangents  at  the  points  A 
and  B  meet  at  D.     If  AB  =  432  feet,  find  the  angle  DAB. 
I  Ans.  6°  29'  35". 

yj      17.    The  radius  of  a  circle  is  1146.28  feet.     Tangents  at  A  and  B  meet 
at  T.     If  AB  =  800  feet,  find  the  angle  TAB  and  the  length  of  A  T. 

Ans.  A  TAB  =  20°  25'  25" ;  A  T  =  426.84. 


18.  The  angle  of  elevation  of  a  point,  P,  from  a  point,  B',  due  west 
of  P  was  found  to  be  B,  and  from  a  point,  A',  due  south  of  B'  it  was 
found  to  be  A .     If  A'Bf  —  c,  show  that  the  height  of  P  was 


Vcot2^.  -cot2 B 


"N 


CHAPTER   IV 


TRIGONOMETRIC   FUNCTIONS   OF  ANY  ANGLE 


II 


X- 


III 


IV 


M 


18.  Coordinates.  Let  XX',  YY'  (Fig.  10)  be  two  lines  inter- 
secting at  right  angles  at  0\  P,  any  point  in  their  plane. 
Draw  PM  perpendicular  to  XX'.  Then,  OM  and  MP  are  called 
the  coordinates  of  the  point  P ;  OM  is  called  the  abscissa,  and 
MP  the  ordinate.  XX'  is  called  the  x-axis,  YY'  the  y-axis. 
Their  intersection,  0,  is  called  the  origin  of  coordinates.  The 
following  conventions  have  been  adopted :  abscissas  measured 
to  the  right  of  the  y-axis  are 
positive;  those  to  the  left, 
negative :  ordinates  meas- 
ured upward  from  the  #-axis 
are  positive ;  downward,  neg- 
ative. Abscissas  are  usually 
denoted  by  the  letter  x,  ordi- 
nates by  y.  With  these  con- 
ventions, any  point  in  a  plane 
can  be  located  with  reference 
to  two  rectangular  axes  in 
the  plane  if  its  coordinates 
are  given.  Thus,  the  equa- 
tions x  =  3,  y  =  4  determine  the  point  whose  abscissa  is  3  and 
whose  ordinate  is  4.  This  point  is  referred  to  as  the  point  (3, 4), 
or,  in  general,  as  the  point  (x,  y),  the  abscissa  being  written  first. 

The  line  joining  P  with  the  origin  is  called  the  distance  of 
P,  and  is  always  considered  positive.  The  axes  divide  the 
angular  magnitude  about  the  origin  into  four  quadrants,  des- 
ignated as  the  first,  second,  third,  fourth,  as  indicated  by  the 
numerals  in  Fig.  10. 

29 


Fig.  10 


30 


PLANE   TRIGONOMETRY 


EXERCISES 

1.  Adopt  a  convenient  scale  and  construct  the  following 
points  :  x  =  A,  y  =  5  ;  x  =  —  3,  y  =  2 ;  x  =  2,  y  =  —  7. 

2.  Construct  the  following  points  :  (3,  5),  (-  2,  -  3),  (3, 0), 
(0,  -  4),  (-  3,  7). 

3.  What  is  the  ordinate  of  any  point  on  the  cr-axis  ?  the 
abscissa  of  any  point  on  the  y-axis  ?  What  are  the  coordi- 
nates of  the  origin  ?  Write  down  the  coordinates  of  three 
points  on  the  ic-axis. 

4.  Find  the  distance  for  each  of  these  points :  (3,  4), 
(-  6,  8),  (15,  -  8). 

19.  Angles  of  any  Magnitude.  In  Chapter  I  only  acute  angles 
were  considered,  and  the  definitions  of  the  trigonometric  func- 
tions there  given  have  no  meaning  except  with  reference  to 


Fig.  n 


acute  angles.  We  shall  now  show  how  angles  of  any  magni- 
tude may  be  generated,  and  shall  distinguish  between  positive 
and  negative  angles  and  give  definitions  of  the  trigonometric 
functions  that  will  apply  to  any  angle. 

If  a  line,  OB  (Fig.  11),  coinciding  at  first  with  OA,  be  made 
to  rotate  about  0  as  a  pivot,  as  indicated  by  the  arrow,  until 
it  comes  into  the  position  OB,  it  is  said  to  describe,  or  generate, 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      31 

or  turn  through,  the  angle  A  OB.  OA  is  called  the  initial  line, 
OB  the  terminal  line,  of  the  angle.  If  OB  revolves  until  it 
comes  again  into  its  original  position,  OA,  the  angle  described 
is  called  a  perigon.  Half  a  perigon  is  called  a  straight  angle  ; 
one  three-hundred-and-sixtieth  of  a  perigon  is  called  a  degree. 
If  OB,  after  describing  the  angle  A  OB,  be  made  to  turn  in  the 
same  way  still  further  through  one  or  more  perigons,  it  will 


.      Fig.  12 

come  back  to  the  same  position,  OB.  In  this  way  may  be 
generated  angles  of  any  magnitude.  Thus,  if  the  angle  indi- 
cated by  arrow  1  (Fig.  12)  be  40°,  the  angle  indicated  by 
arrow  2  is  40°  +  360°,  or  400°,  the  magnitude  of  the  angle  in 
any  case  depending  upon  the  amount  of  turning. 

20.    Negative  Angles.     In  Figs.  11  and  12  we  have  supposed 
OB  to  rotate  in  a  manner  opposite  to  the  hands  of  a  clock 


Fig.  13 

(counter-clockwise).  Angles  may  be  generated  also  by  caus- 
ing OB  to  turn  clockwise,  as  indicated  by  the  arrow  in  Fig.  13. 
Here  we  adopt  a  convention  similar  to  that  in  Sec.  18,  viz.: 
angles  generated  by  a  counter-clockwise  rotation  are  positive ; 
angles  generated  by  a  clockwise  rotation  are  negative. 

Questions.  How  many  degrees  are  there  in  the  negative  angle  described 
by  the  minute  hand  of  a  clock  in  five  minutes  ?  in  three  hours  and  twenty-five 
minutes  ?  How  long,  does  it  take  the  minute  hand  to  describe  the  angle  —  210°  ? 
What  negative  angles  give  the  same  position  of  the  terminal  line  as  the  posi- 
tive angle  150°? 


32 


PLANE    TRIGONOMETRY 


21.    General  Definitions  of  the  Trigonometric  Functions.     Let 

the  initial  line  of  the  angle 
XOB  (Fig.  14)  lie  along  the 
sc-axis  on  the  positive  side 
and  let  P  be  any  point  in 
the  terminal  line ;  denote 
the  angle  XOB  by  A,  and  the 
abscissa,  ordinate,  and  dis- 
tance of  P  by  x,  y,  r  respec- 
tively. Then,  for  every 
position  of  OB,  the  func- 
tions of  the  angle  A  are 
defined  as  follows : 


x    m 


-X 


r 

Fig.  14 


.  .     ordinate 

1.    sine  of  A  equals  — 

distance 

abscissa 


..     a         (( 


2.  cosine 

3.  tangent 

4.  cotangent  "    "       " 

5.  secant         "   "       « 

6.  cosecant     "    "       " 


u    a         (( 


distance 
ordinate 


abscissa 
abscissa 


ordinate 
distance 


abscissa 
distance 


ordinate 


V 

written  sin  A  —- 

r 


cos  A 


tan  A 

=  1£ 

X 

cot  A 

X 

" y 

sec  A 

r 

X 

esc  A 

r 

II 


The  words  ordinate,  abscissa,  distance,  as  used  above,  mean 
the  ordinate,  abscissa,  and  distance  of  any  point  P  on  the 
terminal  line  of  the  angle  considered. 

Let  the  abscissa  of  P  contain  a  units  of  length,  the  ordi- 
nate b  units,  and  the  distance  r  units ;  then,  if  OB  falls  in  the 
first  quadrant,  as  in  Fig.  14,  x  =  +  a,  y  =  +  b,  and  from  the 
definitions  we  have 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      33 


ordinate      b 

Sin  A  =  -zrr— =  - 1 

distance      r 


cos  A 


tan  A 


abscissa  a 
distance  r 
ordinate  _  b 
abscissa      a 


cot  A 


sec  A 


esc  A  = 


abscissa 


ordinate  b 
distance  _  r 
abscissa  a 
distance  _  r 
ordinate      b 


If  the  terminal  line  falls  in  the  second  quadrant  (Fig.  15), 
x  =  —  a,  y  =  +  b,  and  we  have,  remembering  that  the  distance 
is  always  positive, 


sin  A  =  - 1 
r 

cot  A  =  —  7 

0 

—  a 
cos  A  = =  - 

r 

a 

— , 
r 

A                        V 

sec  A  = 

a 

b 
tan  A  = j 

r 
esc  A  =-- 

For  an  angle  in  the  third  quadrant,  i.e.,  an  angle  whose 
terminal   line   lies    in   the 
third  quadrant, 

x  =  —  a,       y  =  —  b. 
For  an  angle  in  the  fourth 
quadrant, 

x  =  -f  a,       y  =  —  b. 


5^P(_a,6) 


X- 


M 


Let  the  student  construct  a 
figure  for  each  of  these  cases  and 
write  down  the  values  of  the  func- 
tions in  terms  of  a  and  6. 

When  the  angle  is  in  the 
first  quadrant,  it  is  readily 
seen  that  the  definitions  given  above  agree  with  those  pre- 
viously given  in  Sec.  4.  The  latter  definitions  are  general, 
and  include  those  of  Sec.  4  as  a  special  case. 


Fig.  15 


34 


PLANE    TRIGONOMETRY 


22.    The  formulas  of   Sec, 
Thus,  from  1  and  2  of  Sec 

1 


4  are  true  also  for  any  angle. 
21, 


smi  =' 

esc  A 

>     esc  ^4  =  — — - ; 
sin  A 

(i) 

from  2  and  5, 

cos  A  = 

sec  A 

sec  A  = ; 

cos  A 

(2) 

from  3  and  4, 

tan  A  =  — — 
cot  ^1 

cot  A  = 

tan  A 

(3) 

Also,  for  any 

position  of  the  revolving  line, 

• 

y 

sin  A       r 

cos  J        X 

V 

-  as  tan  A. 

X 

W 

r 

Similarly,  prove  formula  (5). 

Again,  in  whatever  quadrant  the  terminal  line  may  lie,  the 
abscissa,  ordinate,  and  distance  of  P  form  a  right  triangle, 

Sivillg  2,2  2 

x2  +  y2  =  H. 
Dividing  each  member  of  this  equation  by  r2, 


©■+(;)" 


i. 


sin2  ^4  -J-  cos2  A  =  1. 


(6) 


Prove  formulas  (7)  and  (8)  by  dividing  in  turn  by  x?  and  y~. 

23.  Algebraic  Signs  of  the  Functions.  Let  x  and  y  be  the 
coordinates  of  any  point  in  the  terminal  line  of  an  angle. 
Then,  as  in  Sec.  21,  it  will  be  seen  that  if  the  angle  is  between 
0°  and  90°,  i.e.,  if  the  terminal  line  falls  in  the  first  quadrant, 
x  and  y  are  both  positive ;  and  since  r  is  always  positive,  all 
the  functions  of  the  angle  are  positive. 

If  the  angle  is  between  90°  and  180°,  the  terminal  line  falls 
in  the  second  quadrant,  and  x  is  negative  and  y  positive. 
Therefore,  the  sine  and  the  cosecant  are  positive,  while  all 
the  other  functions  are  negative. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      35 


If  the  angle  is  between  180°  and  270°,  the  terminal  line  is 
in  the  third  quadrant,  and  x  and  y  are  both  negative ;  so  that 
the  sine,  cosine,  secant,  and  cosecant  are  negative,  while  the 
tangent  and  cotangent  are  positive. 

If  the  angle  is  between  270°  and  360°,  the  terminal  line 
falls  in  the  fourth  quadrant,  and  x  is  positive  and  y  negative ; 
so  that  the  sine,  cosecant,  tangent,  and  cotangent  are  negative, 
while  the  cosine  and  secant  are  positive. 

The  student  should  construct  the  figure  to  illustrate  each  case,  as  in  Sec.  21. 

These  results  may  be  arranged  in  a  tabular  form  as  below : 


Quadrant 

SIN 

cos 

TAN 

COT     . 

SEO 

CSC 

I 

+ 

+ 

+ 

+ 

+ 

+ 

II 

+ 

— 

— '< 

— 

- 

+ 

III 

— 

— . 

+ 

+ 

— 

— 

IV 

— 

+ 

— 

— 

+ 

— 

24.    Functions  of  0°,  90° 
Magnitude  of  the  Functions. 


180°,  270°,  360°.     Changes  in  the 
Let  the  variable  angle  A  (Fig.  16) 


B2 

Y 

(o,r) 

%Nx     j^~ 

— «J\  s "i 

JBx 

,    bJ 

fJr^ 

\(r,o) 

(-r,o)\ 

V    ° 

IB 

(o,-r) 

Bi 

Y' 

Fig.  16 


36  PLANE   TRIGONOMETRY 

be  generated  by  a  line,  OB,  of  fixed  length,  r,  lying  at  first 
along  OX  and  revolving  counter-clockwise. 

The  student  should  draw  OB  in  several  positions  in  each  of  the  quadrants. 

1.  Functions  of  0°.  When  A  =  0°,  OB  lies  along  the  a>axis, 
and  the  coordinates  of  B  are  x  =  r,  y  =  0.     Hence, 

.      0  _  ordinate  _  0  _  a_  abscissa  _  r  _ 

distance       r  ordinate      0 

„.      abscissa       r       .  __      distance       r 

COS  0    =  -rr-i =  -  =  1  sec  0    =  — : =  -  =  1, 

distance       r  abscissa       r 

-o      ordinate      0  ^Q      distance       r 

tan  0   =  — — : =  -  =  0,  esc  0   =  — r; — —  =  -  =  oo. 

abscissa       r  ordinate      0 

2.  As  A  increases,  y,  the  ordinate  of  B,  increases ;  its 
abscissa,  x,  decreases,  r  remaining  constant.     Hence,  in  the 

first  quadrant,  sin  A,  -  ?  increases  but  remains  less  than  1 ; 

T  ' 

X  II 

cos  A,    -)    decreases    and    remains   less   than   1;  tan  A,  -> 
r  x 

increases,  being  0  when  A  =  0°,  then  a  number  less  than  1, 
then  1  (when  A  =  45°),  then  a  number  greater  than  1,  which 
can  have  any  value,  however  great,  since  x  gets  nearer  and 
nearer  to  0  as  A  approaches  90°. 

3.  Functions  of  90°.  When  OB  has  turned  through  90° 
(arrow  1),  coming  into  the  position  OB2,  x  =  0,  y  =  r,  and  we 
have 

sin  90°  =  y-  =  -  =  1,  cot  90°  =  -  =  -  =  0, 

r       r  y       r 

cos  90°  =  -  =  -  =  0,  sec  90°  =  -  =  ^  =  oo, 

r       r  x       0 

*  tan  90°  =  ^  =  £  =  oo,  esc  90°  =  -  =  -  =  1. 

x       0  y       r 

*  As  shown  in  2,  if  A,  heing  first  less  than  90°,  be  made  to  approach  90°, 
tan  A  is  positive  and  gets  greater  and  greater,  and  can  be  made  greater  than 
any  assigned  number,  however  great ;  this  is  expressed  by  writing  tan  90°=  +  oo. 
But  if  A  be  greater  than  90°,  and  be  made  to  approach  90°  by  a  clockwise 


n 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      37 

4.  As  A  passes  through  the  value  90°,  x  becomes  negative 

and  increases  *  numerically  while  y  is  positive  and  decreases. 

Hence,  in  the  second  quadrant, 

y 
sin  A,  ->  is  positive  and  decreases; 

T 

cos  A,  -y  is  negative  and  increases; 
r 

tan  A  is  negative  and  decreases ; 
cot  A  is  negative  and  increases ; 
sec  A  is  negative  and  decreases ; 
esc  A  is  positive  and  increases. 

5.  Functions  of  180°.     When  OB  has  turned  through  180° 
(arrow  2),  x  =  —  r,  y  =  0.     Hence, 

sin  180°  =  -  =  0,  cot  180°  =  ^  =  oo, 


r 


0 


cos  180°  =  — -  =  -  1,  sec  180°  =  —  =  -  1, 

r  —  r 

tan  180°  =  —  =  0,  esc  180°  =  £  =  oo. 


r 


0 


6.  When  J.  passes  through  180°,  x  remains  negative  and 
decreases,  y  becomes  negative  and  increases.  Hence,  in  the 
third  quadrant, 

sin  A,  -9  is  negative  and  increases; 

x 
cos  A,  -j  is  negative  and  decreases ; 
r 

V 
tan  A,  -i  is  positive  and  increases. 

rotation,  tan  A  is  negative,,  and  can  be  made  numerically  greater  than  any 
assigned  number,  however  great;  this  is  expressed  by  writing  tan 90°  =  —  oo. 
Hence,  when  it  is  not  known  whether  A  becomes  90°  from  having  been  less 
than  90°  or  from  having  been  greater  than  90°,  there  is  an  ambiguity  in  the 
sign  of  tan  90°,  and  we  must  write  tan90°=±oo.  Similarly,  whenever  the 
symbol  oo  occurs  in  this  discussion  it  should  have  the  double  sign. 

*  In  this  discussion  the  words  increase  and  decrease  refer  to  numerical 
increase  and  decrease. 


38 


PLANE    TRIGONOMETRY 


7.    When  A  =  270°,  x  =  0,  y  =  -  r.     Hence, 


sin  270°  = =  -  1, 


cot  270°  =  —  =  0, 


cos  270°  =  -  =  0, 
r 


tan  270°  = 


0 


sec  270°  =  -  =  cc, 


esc  270° 


=  - 1. 


8.    When  A  passes  through  270°,  x  becomes  positive  and 
increases,  while  y  is  negative  and  decreases.     Hence, 

sin  A,  -•>  is  negative  and  decreases  ; 
r 

cos  A,  -i  is  positive  and  increases; 
r 

y 
tan  .4,  ->  is  negative  and  decreases. 


When  A  =  360°,  OB  has  returned  to  its  original  position 
and  the  functions  of  360°  are  the  same  as  those  of  0°. 

In  the  same  manner,  the  changes  in  the  values  of  cot  A, 
sec  A,  esc  A  may  be  traced  for  the  several  quadrants. 

Questions.  Which  of  the  functions  increase  as  A  increases  from  0°  to  90°? 
Which  decrease?  As  A  increases  from  90°  to  180°  which  of  the  functions 
increase  and  which  decrease  ?  What  is  the  greatest  value  of  sin  A  ?  of  cos  A  ? 
the  least  values  ? 


EXERCISE 

Trace  the  changes  in  the  value  of  sin  A  as  A  varies  from  0° 
to  360°.  Trace  also  the  changes  in  the  values  of  cos  A  and 
tan  A  separately. 

Given  one  Function,  to  find  the  Others.  Suppose  we  have 
given  cot  A  =  f  ^.  Since  cotangent  equals  abscissa  divided  by 
ordinate,  let  us  construct  the  point  B  (Fig.  17)  whose  coordi- 
nates are  60  and  11.     Draw  OB ;   then  the  angle  XOB  has  its 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      39 


and  by  constructing 


cotangent  equal  to  —  •     But  —  =  — :pr 

the  point  £'(—60,  -  11)  we 
determine  the  angle  XOB', 
which  also  has  }$  for  its 
cotangent.  Similarly,  corre- 
sponding to  a  given  value  of 
any  one  function,  there  are 
always  two  positive  angles, 
each  less  than  360°. 

Since  OB=  OB'=  V602+Ha 
=  61,  all  the  functions  of  XOB 
and  XOB'  can  be  written  down. 
Thus,  denoting  XOB  and  XOB'  by  A  and  A '  respectively, 

sin  A  =  i  \,  sec  A  =  f  J,  tan  A '  =  J  J, 

cos  A  =  §£,  esc  A  =  f|,  cot  A'  =  ff, 


tan  ^  =  ft, 

cot  yl  =  f  f , 


sin  yi '  = 


cos  A'  =  —  §£, 


1> 


seo4'=-fl, 
esc  A'  =  —  f|. 


EXERCISES 

Construct  the  two  positive  angles  less  than  360°  correspond- 
ing to  the  given  function  in  each  of  the  following  cases. 
Find  also  the  remaining  functions  of  each  angle. 


1. 

2. 

sin  A  =  }. 

esc  A  —  —  Y-. 

[5.    cot  A  = 7= 

a 

4. 

tan  A  =  ¥9<y. 
cos  ,4  =-}$. 

V5 
6.    tan  ^4  =  -7—  • 

26.  If  05  (Fig.  15)  be  turned,  counter-clockwise  or  clock- 
wise, through  360°  or  any  multiple  of  360°,  it  will  again 
occupy  the  position  shown  in  the  figure.  Hence,  if  an  angle 
be  increased  or  diminished  by  360°  or  any  multiple  of  360°,  its 
functions  are  unchanged. 


40 


PLANE   TRIGONOMETRY 


EXERCISES 

V;1.   tan  496°  =  tan  (360°  +  136°)  =  tan  136°. 

2.    sin  1263°  =  sin  (3  x  360°  +  183°)  =  sin  183°. 
V3.   Prove  cos  762°  =  cos  42°. 

4.  Express  sec  780°  13 '  as  the  secant  of  an  acute  angle. 

5.  Name  three  angles  that   have   the   same  functions  as 
46°  24'. 

1/27.  Functions  of  Negative  Angles.  In  each  of  the  accompany- 
ing figures  let  the  line  OC  turn  counter-clockwise,  generating 
an  angle  XOC  —  A,  indicated  by  arrow  1.  Let  OC  turn  clock- 
wise through  an  angle  equal  in  magnitude  to  XOC,  indicated 
by  arrow  2.     Then  Z  XOC  =  -  A. 


T 

C^Z? 

(-a,b) 

\ 

K        M 

0         J 

V    B'(-a,-b) — 

— 2 

Y' 

X 


Fig.  19 


From  any  point,  B,  in  OC,  draw  BB'  perpendicular  to  XX', 
meeting  OC  in  B'.  In  the  right  triangles  MOB,  MOB'  the 
side  OM  is  common,  and  the  acute  angles  MOB,  MOB1  are 
equal  in  magnitude.  Hence,  the  triangles  are  congruent  (i.e., 
one  could  be  made  to  coincide  with  the  other).  Let  the 
number  of  linear  units  in  OM,  MB  be  a  and  b  respectively, 
and  let  OB  =  OB'  =  r.  Then,  giving  to  a  and  b  their  proper 
signs  according  to  the  quadrants  in  which  OB  and  OB'  lie, 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      41 

the  coordinates  of  B  and  B'  will  be  as  shown  in  the  several 
figures.  Confining  our  attention  to  the  case  in  which  OB  is  in 
the  second  quadrant,  OB'  in  the  third  (Fig.  19),  we  have 


sin  (—  A)  = =  —  sin  A, 


cos  (—  A)  =  — 


cos  A, 


cot  (—  A)  =  -  =  —  cot  A, 

v 

sec  (—  A)  = =  sec  A, 

y  a 


—  b      b  r 

tan  (—  A)  = =  -  =  —  tan^4,       csc  (—  ^4)=  —  - 

v        '      —a     a  '  b 


esc  A. 


X    X* 


The  same  relations  are  true  for  all  cases.  Let  the  student 
prove  them  for  each  figure. 

28.  Functions  of  90°  +  A  in  Terms  of  Functions  of  A.  In  each 
of  the  accompanying  figures  let  the  angle  XOC,  marked  by- 
arrow  1,  be  denoted  by  A.  If  OC  be  drawn  perpendicular  to 
OC,  the  angle  XOC,  marked  by  arrow  2,  will  be  90°  +  A. 
Take  OB  =  OB'  —  r  and  draw  BM,  B'M'  perpendicular  to  XX'. 
In  the  right  triangles  OMB,  OM'B',  OB  =  OB1 'and  the  acute 
angles  BOM  and  OB'M'  are  equal,  having  their  sides  perpen- 
dicular each  to  each.  Hence,  the  triangles  are  congruent. 
Denoting  the  absolute  values  of  OM,  MB  by  a  and  b,  as  in 


42 


PLANE   TRIGONOMETRY 


Sec.  27,  the  coordinates  of  B  and  B'  will  be  as  shown  in  the 
several  figures.     Then,  for  the  case  where  A  falls  in  the  third 


B(-a,bf 

\2 

T 

~<    m' 

v\ 

w 

2L 

Mjy^ 

0 

f^^K-h,-a) 

Y' 

Pig.  23 


quadrant  and  90  +  A  in  the  fourth  (Fig.  24),  we  have 

sin  (90°  +  A)  =  -  -  =  cos  A,        cot  (90°  +  A)  =  -  -  =  -  tan  A, 

cos  (90°  +  ^)  =  -  =  -  sin  A,        sec (90°  +  .4)  =7  =  -  esc  A, 

tan  (90°  +  -4)  =  -  ^  =  -  cot  A,   esc  (90°  +  A)  =  -  -  =  sec  A 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      43 


EXERCISE 

Prove  the  above  relations  for  each  of  the  other  cases. 

From  these  results  it  follows  that  if  an  angle  be  diminished 
by  90°  the  sine  becomes  cosine,  cosine  becomes  negative  sine, 
tangent  becomes  negative  cotangent,  cotangent  becomes  nega- 
tive tangent,  secant  becomes  negative  cosecant,  cosecant 
becomes  secant. 

29.  Functions  of  90°  —  A.  The  figures  of  the  preceding  section 
may  be  used  to  express  the  functions  of  90°  —  A  in  terms  of 
functions  of  A.  In  each  of  those  figures  let  Z.  XOC'=  A,  then 
Z.XOC  =  A  —  90°.     Then  we  have,  as  in  the  last  article, 

sin  (A  -  90°)  =  -  cos  A,  cot  (A  -  90°)  =  -  tan  A, 

cos  (A  -  90°)  =  sin  A,  sec  (A  -  90°)  =  esc  A, 

tan  (A  -  90°)  =  -  cot  A,  esc  (A  -  90°)  =  -  sec  A. 

By  Sec.  27,  if  the  sign  of  an  angle  be  changed,  the  sign  of 
each  of  its  functions,  except  the  cosine  and  the  secant,  will  be 
changed.     Hence, 

sin  (90°  -  A)  =  cos  A,  cot  (90°  -  A)  =  tan  A, 

cos  (90°  -  A)  =  sin  A,  sec  (90°  -  A)  =  esc  A, 

tan  (90°  —  A)  =  cot  A,  esc  (90°  —  A)  =  sec  A. 

These  relations  were  proved  in  Sec.  4  for  the  case  where  A 
is  acute. 

30.  Functions  of  135°,  225°, 
315°.  Let  XOC  (Fig.  26) 
be  an  angle  of  135°.  Then 
Z.COX'  =  45°.  TakeOikf=l, 
and  draw  MB  perpendicular 
to  XX'.  Then  MB  =  1,  OB 
=  V2,  and  the  coordinates 
of  B  are  (—  1,  1)  ;  and  we 
have 


sin  135c 


V2 


cos  135 


44 


PLANE   TRIGONOMETRY 


tan  135°  =  -  1,  sec  135°  =  -  V2, 

cot  135°  =  -  1,  esc  135°  =  V2. 

For  225°  OC  falls  in  the  third  quadrant,  and  for  315°  it 

falls  in  the  fourth ;   in  each  case  take  OM  =  1  and  construct 

the  triangle  OMB  as  above  and  find  the  coordinates  of  B. 

31.    Functions  of  120°,  150°,  210°,  240°,  300°,  330°.      The 

functions  of  each  of  these  angles 

are  closely  related  to  those  of  30° 

and  60°,  as  will  be  seen  from  the 

figures.      We   shall   construct  the 

figure  and  find    the   functions  of 

240°.      Let  XOC  (Fig.  27)  be  an 

angle   of   240°.      Then  the    acute 

angle  X'OC  formed  by  its  terminal 

line  and  the  X-axis  is  60°. 

Construct,  then,  an  angle  of  60° 
in  the  first  quadrant  and  set  off  on 
its  terminal  line  a  distance  OB' =  2, 


0/ 
/f#(l.vSj 


/B  (-1,-VSJ 
V 


Fig.  27 


and  draw  the  perpendicular  B'Mf. 

Then  the  triangle  OB'M'  equals  the  triangle  ABB  of  Fig.  4.  Why  ? 

Hence,  the  coordinates  of  B'  are  1,  v3. 

Now,  setting  off  OB  =  2  and  drawing  the  perpendicular  BMf 

the  triangle  OBM  equals  the  triangle  OB'M',  and  we  have,  for 

the  coordinates  of  B,  —  1,  —  V3.     Hence, 

V3 


sin  240c 


cos  240°  =  - 


tan  240c 


cot240c 


1 

V3 


1 

2 

V3, 


sec  240°  =  -  2, 


esc  240°=- 


2 

V3 


For  each  of  the  other  angles  named  above  construct  first 
the  given  angle,  then  in  the  first  quadrant  an  angle  equal  to 
the  acute  angle  formed  by  the  terminal  line  of  the  given  angle 
and  the  X-axis,  and  proceed  as  above. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      45 


In  the  following  table  are  collected  the  functions  of  the  angles 
named  in  the  caption  of  this  section,  and  for  45°,  30°,  60°  pre- 
viously found.  The  student  should  be  able  readily  to  construct 
the  figure  for  each  case  and  to  obtain  the  results  from  it. 


ANGLE 

SIN 

cos 

TAN 

COT 

SEC 

CSC 

45° 

1 

V2 

1 

V2 

1 

1 

V2 

Vi 

135° 

1 

1 

-   1 

-   1 

-V? 

Vi 

V2 

Vi 

225° 

1 
V% 

1 
Vi 

1 

1 

-  Vi 

-  Vi 

315° 

1 

~  V2 

1 
VI 

-   1 

-1 

Vi 

-  Vi 

1 

V3 

1 

2 

30° 

2 

2 

Vs 

V3 

vi 

2 

V3     ' 

1 

1 

.  •  2 

60° 

2 

2 

Vs 

V3 

2 

V3 

120° 

V3 

1 

-  Vs 

1 

-  2 

2 

2 

2 

V3 

vi 

1 

Vs 

1 

2 

150° 

-  V3 

2 

2 

"  ~Y 

~  Vl 

"  V3 

210° 

1 

VS 

1 

Vs 

2 

-  2 

~  2 

2 

.VI 

V3 

V3 

1 

1 

2 

240° 

V3 

-  2 

2 

~~  2 

vi 

"  V3 

300° 

V3 

1 

-  Vs 

i 

2- 

2 

2 

2 

Vs 

~  V3 

330° 

'     1 

v§ 

l 

-Vi 

2 

-  2 

2 

2 

"  V3 

V3 

Should  it  be  required  to  express  any  of  the  above  functions 
as  a  decimal,  it  is  best  first  to  rationalize  the  denominator 
when  it  contains  a  radical.     Thus, 

1_       V?      1-41421 
V2~    2 


sin  45°  =  — -^  =  -3T-  = 


=  0.70710+. 


H 


46  PLANE    TRIGONOMETRY 

EXERCISES 

•ft4 

1.  Write  down  from  the  above  table  all  the  -functions  in 

which  the  denominator  is  irrational,  and  rationalize  as  above. 

2.  Express  decimally  to  five  places  sin  120°,  sec  30°,  cot  240°. 

32.    By  means  of  formulas  (1)  to  (8)  any  one  of  the  func- 
.  nM      tions  can  be  expressed  in  terms  of  any  one  of  the  others. 
Thus,  to  express  the  tangent  in  terms  of  each  of  the  other 
functions : 

By  formula  (3),  tan  =  — -  > 

which  gives  tangent  in  terms  of  cotangent. 


By  formula  (7), 

tan2  =  sec2  —  1, 

or 

tan  =  Vsec2  —  1. 

By  formula  (8), 

cot  =  Vcsc2  —  1. 

cot       Vcsc2  - 

-1 

By  formula  (4), 

(6), 

sin 

tan  = 

cos 

But,  by  formula 

cos  =  Vl  —  sin2. 

sin 
.  * .  tan  =  — : 

vr 


sin" 


By  formula  (6)  also,        sin  =  Vl  —  cos2. 


Vl  -  cos2 
tan  = 


cos 
Hence,  we  have  the  following  values  for  tangent : 


sm  v  l  —  cos 


VI 


tan  =  — .  = —  =  — -  =  Vsec2  —  1  =     .  =  • 

Vl  -  sin2  cos  cot  Vcsc2  - 1 

Let  the  student  write  down  formulas  (1)  to  (8)  and  by  means  of  them 
express  each  function  in  terms  of  each  of  the  others  separately.. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE      47 

MISCELLANEOUS  EXERCISES 

Prove  the  following  relations : 

1.  tan  x  +  cot  x  =  sec  x  esc  x. 

2.  (tan  x  +  cot  x)2  =  sec2x  +  csc2x. 

3.  cos3 x  tan  x  +  sin2xcosx  tanx  =  sins. 

4.  (1  +  tan  x)  (1  +  cot  x)  =  2  -f  sec  x  esc  x. 

\    e     ..  x      o  secx 

5.  (tan x  —  sin x) csc3x  = 


6. 


1  +  COS  X 

cot  x  —  tan  x      cos  x  +  sin  x 


cos  x  —  sin  x        cos  x  sin  x 

7.  (sin  x  cos  y  +  cos  x  sin  y)2  +  (cos  x  cos  y  —  sin  x  sin  y)2  =  1. 

8.  sin2  x  —  sin2  y  =  cos2  y  —  cos2  x. 

9.  cos2x.—  sin2 y  =  cos2y  —  sin2x. 

Determine  x  from  the  following  equations,  x  being  an  angle  between 
0°  and  90° : 

10.  sin  6  x  =  £.  [One  solution  is  given  by  the  equation  6  z  =  30°,  another 
by  the  equation  6x=150°;  obtain  the  others  by  adding  360°  successively  to 
these  angles.] 

11.  sin  3  x  =  cos  x.  Ans.  x  =  22£°,  or  45°. 

12.  tan  6  x  =  cot  3  x.  Ans.  x  =  10°. 
J    13.    5  tan2x  +  3  sec2x  =  11.  Ans.  x  =  45°. 

14.  versin  2  x  =  f .  Ans.  x  =  60°. 

15.  tan  5x  =  1.  Ans.  x  =  9°,  45°,  or  81°. 

V3 

16.  sin  5  x  =  — -  •  Ans.  x  =  12°,  24°,  or  84° 

2 


\ 


CHAPTER   V 


ADDITION  AND   SUBTRACTION   FORMULAS 


33.    To  prove  the  relation 

sin  (x  +  y)  =  sin  x  cos  y  +  cos  x  sin  y, 

or,  the  sine  of  the  sum  of  two  angles  equals  the  sine  of  the 
first  times  the  cosine  of  the  second,  plus  the  cosine  of  the  first 
times  the  sine  of  the  second. 

We  shall  first  prove  this  and  the  succeeding  propositions  for 
the  case  where  x  and  y  are  each  less 
than  90°,  and  their  sum  also  less  than 
90°.  In  Fig.  28,  let  Z  LON  =  x, 
Z  NOM  =  y,  then  Z  LOM  =  x  +  y. 
Prom  any  point,  C,  in  the  terminal 
line  of  x  +  y  draw  CA  and  CD  per- 
pendicular to  OL  and  ON  respectively ; 
DB  and  DE  perpendicular  to  OL  and 
AC  respectively.  The  acute  angles 
LON  and  DCE  have  their  sides  per- 
pendicular  each   to  each. 

.\ZDCE  =  x. 

K      AC      BD  +  EC 
Now,     sin  (x  +  y)  =  — 

BD 
OC 
EC 


OC 


OC 


EC 

OC 


But 


and 


BD       OD 

—  = x  — -  =  sin  x  cos  y, 

OD       OC  " 

EC       CD 

—  = X =  cos  x  sin  y. 

OC       CD       OC  y 


Substituting  these  values  above,  we  have 

sin  (x  -f  y)  =  sin  x  cos  y  +  cos  x  sin  y. 

48 


(9) 


ADDITION    AND   SUBTRACTION    FORMULAS         49 

7?  7) 
In  deciding  upon  the  artifice  by  which  — —  is  to  be  transformed  into 

sin  x  cos  y,  we  may  proceed  as  follows  : 

_-.  BD     BD     •"    ' 

First  write  __=__x_. 

Then,  under  BD,  write  the  hypotenuse  of  the  right  triangle  of  which  BD 
is  a  leg,  that  is,  OD ;  then,  of  course,  OD  must  also  be  written  over  OC,  giving 
BD=BD     OD, 
OC      0DX  OC' 
So  in  each  of  the  similar  cases  below. 

34.  To  prove  the  relation 

cos  (x  -\-  y)  =  cos  x  cos  y  —  sin  x  sin  y, 
or,  the  cosine  of  the  sum  of  two  angles  equals  the  cosine  of 
the  first  times  the  cosine  of  the  second,  minus  the  sine  of  the 
first  times  the  sine  of  the  second. 

From  Fig.  28, 

cos(x  +  y)  =  —  = 

I- 

A  ED 

and  ScT 

Substituting  these  values  above, 

cos  (x  +  y)  =  cos  x  cos  y  —  sin  x  sin  y.  (10) 

35.  To  prove  the  relation 

sin  (x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y, 
or,  the  sine  of  the  difference  of  two  angles  equals  the  sine  of 
the  first  times  the  cosine  of  the  second,  minus  the  cosine  of  the 
first  times  the  sine  of  the  second. 

In  Fig.  29,  let     Z  LON  =  x,  Z  NOM  =  y. 

Then,  ZLOM=x-y. 

From  any  point,  C,  in  the  terminal  line  of  x  —  y  draw  CA, 
CD   perpendicular   to    OL    and    ON   respectively,    as    above. 


OB-  ED 

OB       ED 

OC 

o~c~~oc' 

OB       OD  _ 
ODX  OC~ 

cos  x  cos  y, 

ED      CD  _ 
CDX  OC~ 

sin  x  sin  y. 

50 


PLANE   TRIGONOMETRY 


Draw  also  DB,  CE  perpendicular  respectively  to  OL,  DB.    The 
acute  angles  LON,  CDE  are  equal.     Why? 


s      AC      BD- 


BD 
OC 


ED 

oc' 


Now, 
and 


H 


BD       OP 
OD  X  OC 


sm  x  cos  y, 


ED       CD 
~CDX  OC 


(11) 


BD 
OC 
ED 

OC       CD       OC  y 

Hence,       sin  (x  —  y)  =  sin  x  ces  y  —  cos  x  sin  y. 
36.   To  prove  the  relation 

cos  (x  —  y)  =  cos  x  cos  y  +  sin  x  sin  y, 
or,  the  cosine  of  the  difference  of  two  angles  equals  the  cosine 
of  the  first  times  the  cosine  of  the  second,  plus  the  sine  of  the 
first  times  the  sine  of  the  second. 
From  Fig.  29, 
cos  (x  —  y) 

Now, 

and 


OA  _ 
OC~ 

OB  +  EC 

OC 

OB 
OC 

OC 

OB  _ 
OC~ 

OB 
^X 

OD 
OC" 

cos  x  cos  y, 

EC 

EC 

CD 

OC       CD       OC  J 

cos  (x  —  y)  =  cos  x  cos  y  +  sin  x  sin  y. 


(12) 


ADDITION   AND   SUBTRACTION   FORMULAS        51 


In  formulas  (9),  (10),  (11),  (12),  it  should  be  noted  that  for  the  sine  the 
algebraic  sign  between  the  two  terms  on  the  right  is  the  same  as  that  between 
x  and  y  on  the  left ;  for  the  cosine  the  algebraic  signs  are  different  in  the  two 
members. 

Formulas  (9)  and  (10)  are  called  addition  formulas,  (11) 
and  (12)  subtraction  formulas. 

37.  General  Proof  of  the  Addition  and  Subtraction  Formulas. 
By  paying  attention  to  algebraic  signs,  we  may  apply  the 
proof  given  in  Sees.  33,  34  to  the  case  where  x  and  y  are 
acute  angles  whose  sum  is  between  90°  and  180°.  In  this 
case  the  terminal  line  of  the  angle  x  +  y  falls  in  the  second 
quadrant. 

F 


r 


AM 


In  Fig.  30,  let  Z  XON  =  x,  Z  NOM  =  y. 

Then,  Z  XOM  =  x  +  y. 

From  any  point,  C,  in  the  terminal  line  of  x  +  y  draw  the 
perpendiculars  CA  and  CD  to  XX'  and  ON  respectively ;  also 
DEy  DB  perpendicular  respectively  to  CA,  XX'. 

Then,  Z  DCE  =  x. 

Let  the  coordinates  of  C  be  (—  m,  n),  of  D  (a,  b),  and  the 
number  of  linear  units  in  ED  and  OC  be  I  and  r  respectively. 


52  PLANE   TRIGONOMETRY 

N     n       AC       BD  +  EC       BD       EC 
Now,  Bin(*  +  y)  =  -  =  —  =        QC       =  7^  +  7^' 

BD       BD       OD 
But  —  =  —  X—  =8111*  cosy, 

EC       EC       CD 
and  —  =  ^X  —  =  C0B*smy. 

.*.  sin  (a;  +  y)  =  sin  a;  cos  y  -f-  cos  a?  sin  y.       (a) 

—  m 
Also,  cos  (x  +  y)  = 

But,  from  the  figure,  I  =  m  +  a. 

.'.  —  m  =  a  —  I. 

—  m       a  —  I       a       I       OB       ED 

.'.  cos(x  +  y)  = = = =  777;  —  7777* 

v         VJ         r  r  r       r       OC       OC 

OB       OB       OD 
But  —  =  —  X— =  cos*cosy, 

ED       ED      CD 
and  —  -  —  X  —  =  sin*  sin  y. 

'  V.  cos  (x  4-  y)  =  cos  *  cos  y  —  sin  *  siny.       (b) 

From  (a)  and  (b)  it  follows  that  the  addition  formulas  are 
true  when  x  and  y  are  each  less  than  90°  and  their  sum  greater 
than  90°.  The  proof  is  extended  so  as  to  include  all  possible 
cases,  as  follows  : 

1.  Let  one  of  the  component  angles,  say  x,  be  increased  by 
90°,  and  let  90°  +  x  =  A. 

Then,  sin  (.4  +  y)  =  sin  (90°  +  x  +  y) 

=  cos  (x  +  y)  by  Sec.  28 

=  cos  x  cos  y  —  sin  x  sin  y. 
But  again,  by  Sec.  28, 

sin  A  =  cos  x, 
cos  A  =—  sin  x. 


ADDITION   AND   SUBTRACTION   FORMULAS         53 

Substituting  these  values  for  cos  x  and  —  sin  x,  we  have 

sin  (A  -f-  y)  =  sin  A  cos  y  -f  cos  A  sin  y.  (c) 

Similarly,      cos  (A  -f  y)  =  cos  (90°  +  x  +  y) 
=  —  sin  (x  +  y) 
=  —  sin  #  cos  ?/  —  cos  cc  sin  y ; 
or  cos  (^4  -f  y)  =  cos  A  cos  y  —  sin  A  sin  y.  (d) 

From  (c)  and  (d)  it  follows  that  the  addition  formulas  are 
true  after  one  of  the  acute  angles  x  or  y  has  been  increased 
by  90°. 

2.    Next  let  y  also  be  increased  by  90°,  and  let 

90°  +  y  =  B. 
Then,  sin  (A  +  B)  =  sin  (A  +  90°  -f  y) 

=  cos  (A  +  y) 

=  cos  A  cos  y  —  sin  A  sin  y.    By  (d) 
But  sin  B  =  cos  y, 

and  ,  cos  B  =—  sin  ?/. 

.'.  sin  (.4  +  B)  =  sin  ^4  cos  i?  +  cos  ^4  sin  B.         (e) 
Similarly,      cos  {A  +  B)  =  cos  (.4  +  90°  +  y) 
=  —  sin  (.4  +  y) 

—  —  sin  A  cos  y  —  cos  A  siny,    by  (c) 
or  •  cos  {A  +  B)  =  cos  ^4  cos  B  —  sin  ^4  sin  B.        (f) 

From  (e)  and  (f )  it  follows  that  the  addition  formulas  are 
true  after  both  of  the  component  angles  have  been  increased 
by  90°.  By  successive  applications  of  the  same  course  of 
reasoning  it  may  be  shown  that  they  are  true  after  one  or 
both  of  the  component  angles  have  been  increased  by  any 
number  of  times  90°. 

[Thus,  for  the  next  step,  add  90°  to  A,  and  let  90°  +  A=A', 
and  proceed  as  before.] 

They  are  true,  therefore,  for  any  positive  values  of  the 
component  angles. 


54  PLANE   TRIGONOMETRY     • 

In  exactly  the  same  way  the  subtraction  formulas  (11)  and 
(12)  may  be  extended  so  as  to  include  all  positive  values  of  x 
and  y. 

Now  let  one  of  the  component  angles,  say  y,  be  negative. 
Let  y  =—  B  where  B  is  positive ;  then, 

sin  (x  4-  y)  =  sin  (x  —  B)  =  sin  x  cos  B  —  cos  x  sin  B, 

by  formula  (11). 
.*.  sin  (x  +  y)  =  sin  x  cos  y  +  cos  x  sin  y, 
since  cos  B  =  cos  y  and  —  sin  B  =  sin  y. 

Therefore,  formula  (9)  is  true  when  one  of  the  angles  is 
negative. 

Next  let  both  x  and  y  be  negative  and  let  x  =  —  A,  y  =  —  B, 
where  A  and  B  are  positive. 

Then,     sin  (x  -f  y)  =  sin  (—  ^4  —  5)  =  —  sin  (J.  +  B) 
—  —  sin  A  cos  i*  —  cos  A  sin  5. 
.'.  sin(#  +  y)  =  sin  a:  cos  y  -f-  cos  #  sin  3/. 

Therefore,  formula  (9)  is  true  when  both  the  component 
angles  are  negative. 

In  the  same  way  formulas  (10),  (11),  (12)  may  be  extended 

so  as  to  include  the  case  where  one  or  both  the  component 

angles  are  negative.     The  addition  and  subtraction  formulas 

are  therefore  true  universally. 

Let  the  student  give  the  proof  of  formulas  (10),  (11),  (12) :  first,  when  x  is 
positive  and  y  =  —  B;  second,  when  x  =  —  A,  y  =  —  B. 

38.    To  prove  the  relation 

tan  x  +  tan  y 

tan  (x  +  y)  = ^        y   .  (13) 

„_  ..  v  ;      1  —  tan  x  tan  y  v     ' 

We  have 

sin  (x  -f-  v)      sin  x  cos  y  -f-  cos  x  sin  y 

tan (x  -4-  v)  = ^ = — — 

^  '      cos  (cc  +  2/)      cos  x  cos  y  —  sin  x  sin  y 

sin  x  cos  ?/  cos  cc  sin  y 

cos  «  cos  y  cos  #  cos  y  _    tan  x  4-  tan  y 

~  cos  cc  cos  y  sin  cc  sin  ?/      1  —  tan  x  tan  y 

cos  #  cos  2/  cos  x  cos  2/ 


ADDITION   AND   SUBTRACTION   FORMULAS         55 

We  may  proceed  in  the  same  way  and  deduce  a  formula  for 

tan  (x  —  y) ;   or  we  may  substitute  —  y  for  y  in  formula  (13). 

Thus, 

/  v  r     .  /       x  -i       tan  x  +  tan  (—  y) 

tan  (x  —  y)  =  tan  Vx+(—y)l= — v-     y\, 

v         ^y  L         v     ^J      1  -tanartan(— y) 

.  tan  x  — tan  y  _.. 

or  tan  (x  —  y)  = — .  (14) 

7       1  +  tan  x  tan  y  v     ' 

Express  formulas  (13)  and  (14)  in  words. 

Similarly,  let  the  student  prove 

.  ,  x       cot  x  cot  y  —  1 

cot  (a;  +  y)  = ; > 

v         9/        cot  y  +.  cot  x 

,  ,  /  s      c°t  *  cot  y  -4-  1 

and  cot  (x  —  y)=  — - ^— 

v  7        cot  y  —  cot  as 

39.    To  prove  the  relations 

sin  A  +  sin  B  =  2  sin  \  (A  +  B)  cos  £  (A  —  B),  (15) 

sin  A  —  sin  B  =  2  cos  ^  (A  +  B)  sin  \  (A  —  B),  (16) 

cos  A  +  cos  B  =  2  cos  \  (A  +  B)  cos  \  (A  —  B),  (17) 

cos  A  —  cos  B  =  —  2  sin  \  (A  +  B)  sin  £  (A  —  B).         (18) 
Let  ^4 .  =  x  +  y,  B  =  x  —  y. 

Then,  *  =  $(,!+£),      y  =  J(^-*)- 

By  the  addition  and  subtraction  formulas, 

sin  A  =  sin  x  cos  y  +  cos  cc  sin  y, 
sin  2?  sh  sin  x  cos  y  —  cos  x  sin  y. 
Adding, 

sin  A  -4-  sin  5  =  2  sin  #  cos  y 

=  2sin^(^  +£)cos£(,4  _  B). 
Subtracting, 

sin  A  —  sin  B  =  2  cos  x  sin  ?/ 

=  2  cos  i(.4  +  J5)sin  ^(^  -  B). 


56  PLANE    TRIGONOMETRY 

Similarly,  cos  A  =  cos  x  cos  y  —  sin  x  sin  y, 

cos  B  =  cos  x  cos  ?/  +  sin  x  sin  ?/. 
Adding,  cos  A  +  cos  i?  =  2  cos  x  cos  ?/ 

=  2  cos  \{A  +  J3)cos  ^(.4  -  5). 

Subtract  and  prove  formula  (18). 

Formula  (15)  is  equivalent  to  the  general  proposition :  the 
sine  of  one  angle  plus  the  sine  of  another  equals  twice  the  sine 
of  half  their  sum,  times  the  cosine  of  half  their  difference. 

Express  formulas  (16),  (17),  and  (18)  similarly.  From 
formulas  (15)  and  (16),  by  division,  prove 

sin  A  +  sin  B  _  tan  \(k  +  B) 
sin  A  —  sin  B  ~~  tan  }  (A  —  B)  ' 

40.  Functions  of  Twice  any  Angle.  In  formula  (9)  let  y  =  x, 
and  we  have 

sin  (x  +  x)  =  sin  x  cos  x  +  cos  x  sin  x, 

or  sin  2  x  =  2  sin  x  cos  x.  (20) 

Let  y  =  x  in  formula  (10),  and  we  have 

cos  (x  +  x)  =  cos  x  cos  x  —  sin  x  sin  x, 
or  cos  2  x  =  cos2  x  —  sin2  x.  (21) 

Since  sin2 a;  +  cos2 a;  =  1,  we  have  also 

cos  2  x  =  1  —  sin2£c  —  sin2«, 
or  cos  2  x  =  1  —  2  sin2x ;  (22) 

and  cos  2  x  =  cos2rr  —  (1  —  cos2a?), 

or  cos  2  x  =  2  cos2  x  -  1.  (23) 

Formula  (20)  may  be  read:    the  sine  of  twice  any  angle 
equals  twice  the  sine  of  the  angle  times  the  cosine  of  the  angle. 
Express  formulas  (21),  (22),  and  (23)  similarly. 
Put  y  =  x  in  formula  (13)  and  prove 

2tanx  /0.x 

tan2x  = (24) 

l-tan2x  v     } 


\ 


ADDITION   AND   SUBTRACTION   FORMULAS         57 

41.    In  formulas  (22)  and  (23),  let  2x  =  A,  then  x  =  \  A, 
and  we  have 

cos  A  =  1  —  2  sin2  £  ^4, 

2  sin2 }  A  =  1  -  cos  A  j  (25) 

cos  A  =  2  cos2  %A—1, 

or  2  cos2  ^  A  =  1  +  cos  A.  (26) 

From  formulas  (25)  and  (26),  if  cos  A  be  given,  sin  £  J.  and 
cos  |  ^4  can  be  found. 

From  formulas  (25)  and  (26),  we  have,  by  division  and 
reduction, 

/l  —  cos  A 


^     1.    Given  cos  30°  =  ^ 

Putting  A  =  30°  in  formula  (25), 


EXERCISES 

?  find  sin  15°  and  cos  15c 


=  1 -  =  - ?. 

2  2 

.-.sin  15°  =  i  V2  -  V3. 
Similarly,  from  formula  (26), 

2  cos2 15°=  1+—, 

2 


cosl5°  =  £  V2+  V3. 


2.  Given  cos  36°  =  +  *,  prove  cos  18°  =  \J5  +V^  ■ 
Find  also  sin  18°.  8 

3.  Find  sine,  cosine,  and  tangent  of  22°  30'. 

42.  The  addition  and  subtraction  formulas  may  be  used  to 
express  the  sine  and  cosine  of  an  angle  in  terms  of  functions 
of  its  supplement.     Thus, 

sin  (180°  -  A)  =  sin  180°  cos  A  -  cos  180°  sin  A. 


W 
f^ 


58  PLANE   TRIGONOMETRY 

Since  sin  180°  =  0, 

and  cos  180°  =  —  1, 

this  becomes  sin  (180°  —  A)  =  sin  A.  (28) 

Similarly,  cos  (180°  —  A)  =  —  cos  A.  %       (29) 

Since  A  is  the  supplement  of  180°  —  A,  we  have  the  impor- 
tant relations  :  the  sine  of  an  angle  equals  the  sine  of  its  sup- 
plement ;  the  cosine  of  an  angle  equals  the  negative  cosine  of 
its  supplement.     • 

Similar  relations  can  be  proved  for  the  other  functions. 

Thus,  sec  (180°  -A)  = — ^ = — -  =  -  sec  A. 

y      cos  (180  —  A)       —  cos  A 

Similarly, 

esc  (180°-  A)=cscA, 

a.      /iQfto       A,      sin  (1&0° -A)        sin  A 

tan  (180  -  A)  =  J^  _  A)  =  ^^j  =  -  tan  A, 

cot  (180°- 4)=  -cot  4. 
Similarly, 

sin  (270°  -A)  =  -cosA,  etc. 

The  addition  and  subtraction  formulas  may  be  used. also  to 
recall  the  values  of  sin (90°  +  A)  and  cos  (90°  +  A)  in  terms 
of  functions  of  A  as  given  in  Sec.  28. 

Thus,  sin  (90°  +  A)  =  sin  90°  cos  A  +  cos  90°  sin  A  =  cos  A, 
since  sin  90°  =  1  and  cos  90°  =  0. 

Also,    cos  (90°  +  A)  =  cos  90°  cos  A  —  sin  90°  sin  A  =  —  sin  A. 

EXERCISES 

1.  Express  the  functions  of  180°  —  A  in  terms  of  functions 
of  A  by  means  of  a  figure,  as  in  Sec.  28,  taking  A  in  the  first 

,/ /quadrant. 

2.  Find    an   angle   tjzat   has   the    same  sine  as   (a)  150°; 
/(b)  123°  22' 36";    (cV^ 

3.  If  cos  18°  =  " 


ADDITION   AND   SUBTRACTION   FORMULAS        59 

MISCELLANEOUS  EXERCISES 
"Pj-"vft  the  following  relations  : 

sin  (x  +  y)  _  cot  y  -f  cot  x  _l-cos^l 


2. 


sin  (x  —  y)      cot  y  —  cot  x  sin  ^L 

sin  (x  —  y)  _   cot  2/  —  cot  x  sjn  ^ 

cos  (x  +  y)      cot  ?/  cot  x  —  1  '    versin  A  ~  C°         ' 


r      1/      3     cotx+1  =  sec2x  +  tan2x.  9.    tan  \A  +  cot£J.  =  2  esc  A 

/Y^r  cot  x  —  1 

it"  ^         x,  x       1-tanxtany  1A      .    ,  A        .  /versin  A 

»f>  4.    cot(x  +  y)  = -•  10.    sin£J.=\/ 

tan  x  +  tan  y  *        2 

•     ,   a  ll-cosA  ....        2  tan  i  J.  . 

5.    sini^-x/ 11. -  =  smi. 

7  \         2  1  +  tan2  4  ^1 


a* 


1+cos^i                   __     1- tan2 -J- J. 
cos£^.=\ « 12-    -^— =  cosJ.. 


4  0     sin  2  x  +  sin  2  y 

13. —  tan  (x  +  y). 

cos2x  +  cos2y 


2  l  +  tan2£J. 


,  .     cos  2  x  +  cos  2  y 

14.  -  — ^  =  -cot(x  +  y)cot(x-y). 
cos2x-cos2y  v  •       v        y/ 

l  ' 
,  _     sin  7  x  +  sin  x 

15.    =  tan  4  x. 

cos  7  x  +  cos  x  * 

.  .     -  cos  52°  30'  +  cos  7°  30'        ^r 

16.  Prove  =  v3. 

sin  52°  30'  +  sin  7°  30' 

17.  Prove  sin  3  x  =  3  sin  x  —  4  sin3x. 

Write  sin  3  x  =  sin  (2x  +  x) ;  apply  formulas  (9),  (20),  (22). 
1.8.    Express  cos  3  x  in  terms  of  cos  x. 

1Q     -d  .  4tanx  — 4tan3x 

19.  Prove  tan  4  x  = 

1  —  6  tan2x  +  tan4x 

20.  Prove  sin  (A  +  B  4-  C)  =  sin  A  cos  B  cos  C  +  cos  A  sin  B  cos  C 

+  cos  A  cos  B  sin  O  —  sin  A  sin  J5  sin  C. 
Put  x  =  J.  +  5,  y  =  Cin  formula  (9). 

21.  Prove  cos  (A  +  B  +  C)  =  cosA  cos  B  cos  C  -  sin  J.  sin  JS  cos  C 

—  sin  ,4  cos  B  sin  C  —  cos  A  sin  5  sin  C 


60  PLANE    TRIGONOMETRY 

22.    Use  the  results  of  Exercises  20  and  21  to  prove 

tan  A  +  tan  B  +  tan  C  —  tan  A  tan  B  tan  C 
tan  A  tan  B  —  tan  A  tan  C  —  tan  B  tan  (7 


tan(^i  +  5+  C)  = 


23.  If  -4,  5,  C  are  the  angles  of  a  triangle,  prove,  by  using  the  result 
of  Exercise  22,  that 

tan  A  +  tan  B  +  tan  C  =  tan  A  tan  B  tan  C. 

24.  Express  sin  3  a:  in  terms  of  sinx  by  putting  A  =  B  =  C  =  x  in 
Exercise  20. 

25.  Express  cos  3  x  in  terms  of  cosx  by  using  the  result  of  Exercise  21. 

]/ no    -d  •    o  2  tanas 

*   26.    Prove  sm2i  = — 

1  +  tan2x. 


/ 


27.   Prove  tan  £x  = 


28.   Prove 


1  +  cosx 
1  —  tan  x      cot  x  —  1 


1  +  tan  x      cot  x  +  1 

v     ««     ^  sin  #  1  +  cos  x 

29.  Prove 1 ^—  =  2  esc  x. 

1  +  cosx  sinx 

__     _  .    _         sin2  2  x  — sin2  x 

30.  Prove  sin  3  x  = 


31.   Prove^-?^  =  2cos2x  +  l. 
sinx 

*  32.   Prove  sin  50°  +  sin  10°  =  sin  70°. 

33.    Prove  cos4x  —  sin4x  =  cos  2  x. 

/_.     _  sin3x  +  sin4x  .  , 

/  34.    Prove ! =  cot  \  x. 

cos  3  x  —  cos  4  x 

OK     _,  sin  50°  + sin  30°  .  KAO 

35.  Prove =  cot  50°. 

cos  50°  + cos  30° 

M     -r,         sin  3  x  —  sin  x 

36.  Prove =  tan  x. 

cos  3  x  +  cos  x 

*  37.   Prove  sin  15°  =  cos  75°  =  \  (V6  -  V5). 

38.  Prove  cos  15°  =  sin  75°  =  \  ( V6  +  V2~). 

39.  Prove  tan  15°  =  cot  75°  =  2  -  V3. 


40. 

41. 

1/42. 

43. 

4  44. 

45. 


PLANE    TRIGONOMETRY 

Prove  cot  15°  =  tan  75°  =  2  +  Vs. 


60  a 


Prove  sin 22|°  =  |  V2  -  V2  ;  cos 22£°  =  i  V2  +  V2. 
Prove  tan  22i°  =  V2  -  1 ;  cot  22 1°  =  V2  +  1. 
Prove  (sin  \x  ±  cos  £  x)2  =  1  ±  sinx. 
Prove  1  +  tan  x  tan  \  x  =  sec  x. 

-n  /1  1    w      2(1  + sinx) 

Prove  (1  +  tan£x)2  =  — — ! '-. 

1  +  cos  x 


46.    Prove 


cos  x  —  sin  x 


cos2x 


cos  x  +  sin  x      1  +  sin  2  x 


47. 

1^8. 
«49. 


Prove  sec  2  x  +  tan  2  x  +  1  = 

1  —  tan  x 

Prove  ( Vl  +  sin x  ±  Vl  —  sin  xj=  4 sin2  \ x,  or  4  cos2  | x. 
Prove  tan  (45°  4-  x\-  tan  (45°  —  x)  =  2  tan  2  x.     


50. 
51. 
52. 

^J53. 

54. 
J55. 
56. 

57. 
58. 
59. 

n>60. 
61. 
62. 


Prove  cot  (45°  —  x)  —  tan  (45°  —  x)  =  2  tan  2  x. 

Prove  sec2  x  esc2  x  =  sec2x  +  csc2x  ;  tan  (60°  +  x)—  cot  (30°  —  x)  =  0. 

Prove  sin  (x  +  y)  cos  (x  —  y)  —  cos  (x  +  y)  sin  (x  —  y)  =  sin  2  y. 


cos3x      sin3x 
Prove  — - — -  -\ —  =  2  cot  2  x. 


Prove  (sin  x  —  sin  y)2  +  (cos  x  —  cos  y)2  =  4  sin2 . 

Prove  (sinx  +  cosx)(2  —  sin2x)  =  2(sin8x  +  cos3x). 

-n  -91/^1         i\9     -.       •         sin3x  — cos3x     «  .    ^ 

Prove  sin2  \x  (cot  \  x  — 1)2=  1—  sin x ; =  2  sin  2x  —  1. 

sin  x  +  cos  x 

Prove  sin(x-y)  +  sin(y~g)  +  sin(z-x)  =  0 
cos  x  cos  y       cos  y  cos  z       cos  x  cos  z 

Sin  (45  +  x)  sin  (45  —  x)  =  $  cos  2  x. 


1  + 


4  tan2  x 


(cos2  x  -  sin2  x)2  "   (1  -  tan2  x)2 

sec  x  tan  x  —  sin  x 


1  +  cos  x  sin3  x 

(asinx  +  ftcosx)2  +  (a  cos  x  —  b  sin  x)2  =  a2  +  ft2. 

£  +  cosx  =  2  esc (30  +  -J cos (30  -  -J . 


^h 


•f- 


y 


<QOb  PLANE   TRIGONOMETRY 

If  A  +  B  +  C  =  180°,  show  that 

63.'  Cot  I A  +  cot  £  B  +  cot  J  C  =  cot  \  A  cot  £  £  cot  |  C. 
1(54.    Sin  ^1  +  sin  5  +  sin  C  =  4  cos  J  ^4  cos  1 5  cos  £  C. 

65.  Cos^.  +  cos.6  +  cosC—  1  =  4  sin  £  ^.  sin  £2*  sin  £  C. 

66.  Sin  2  J.  +  sin  2  B  +  sin  2  C  =  4  sin  J.  sin  5  sin  C. 

67.  Sin  2  ^.  +  sin  2  B  —  sin  2  C  =  4  cos  J.  cos  B  sin  (7. 

68.  Sin2  A  +  sin2  5  +  sin2  C  =  2  +  2cosA  cosB cos  C. 

Solve  the  following  equations  : 
^-^69.    Sin2x  =  2cosx. 

70.  Sin  x  +  cos  x  =  sec  x. 

71.  4cosx  =  3secx. 

72.  Cos  2  x  —  sin  x  =  | . 
^-—^73.   3  sin  x  —  2  cos2  x  =  0. 

74.  Given  rsinx  =  a,  r  cosx  =  6,  show  that  tanx  =  -  and  r  — 

,  b  sin  x 

cosx 

75.  Given    r  cosx  cosy  =  a,    r  sin x  cos y  =  6,    rsinx  =  c,    show   that 

6         .   ^  ccosx      csinw 

tan  x  =  -  and  tan  y  = = -  • 

a  a  b 

- — >J76.    Solve  a  sin  x  4-  6  cosx  =  c.    Let  rasinz  =  a  and  »»cos2-  6,  then 
show  that  m  =  Va2  +  b2  and  m  cos  (x  —  z)  =  c. 

77.   Given  74  a  =  4,  b  —  3,  to  find  x  and  r. 


\2/i 


CHAPTER   VI 


PROPERTIES   OF   TRIANGLES;    SOLUTION   OF   OBLIQUE 
TRIANGLES 

43.  In  the  solution  of  oblique  triangles  four  cases  present 
themselves : 

1.  Given  a  side  and  two  angles. 

2.  Given  two  sides  and  their  included  angle. 

3.  Given  two  sides  and  an  angle  opposite  one  of  them. 

4.  Given  the  three  sides. 

The  solution  is  effected,  as  in  the  case  of  right  triangles, 
by  means  of  relations  between  the  sides  and  the  angles. 
Formulas  embodying  such  relations  will  now  be  established. 

44.  Law  of  Sines.  In  any  triangle  the  sides  are  propor- 
tional to  the  sines  of  the  opposite  angles. 


I) 


Fig.  31 


Fig.  32 


Proof.     In  the  triangle  ABC  draw  the  perpendicular  CD. 
Then,  if  all  the  angles  are  acute,  as  in  Fig.  31,  we  have 


.'.by  division, 


•            A               CD 

sin  A  =  — —  > 

0 

n       CD 

sin  B  = 

a 

CD 

sin  A         b 

a 

— —  S3 

— 

sin  B       CD 

b 

a 

61 


62  PLANE   TRIGONOMETRY 

If  one  of  the  angles,  as  B  (Fig.  32),  be  obtuse,  then,  since 
the  sine  of  an  angle  equals  the  sine  of  its  supplement, 

CD 


sin  B  = 

sin 

CBD  = 

a 

lile 

sin  A  = 

CD 
b 

?  as  before. 

sin^l 

a 

"  sin  B 

T 

In  like  manner, 

sin  A 
sin  C 
sin  B 
sin  C 

a 

—  > 
c 

b 
c 

These  results  may 

be  written 
a             b 

c 

sin  A       sin 

B~ 

sin  C 

(30) 


45.    Case  1.    Given  one  side  and  two  angles. 
Given  a  =  263,  A  =  81°,  B  =  52°,  to  find  b,  c,  C. 

The  law  of  sines  and  the  relation  A  +  B  +  C  =  180°  suffice 
for  the  solution. 

We  have  at  once  C  =  180°  -(A+B)  =  47°. 

To  find  b,  apply  the  law  of  sines,  preferably  writing  the 
unknown,  b,  first.      Thus, 

b  _  sin  B 
a      sin  A 
.-.  log  b  —  log  a  =  log  sin  B  —  log  sin  A, 
or  log  6  =  log  a  +  log  sin  B  —  log  sin  A. 

log  263=    2.41996 
log  sin  52°  =    9.89653  -  10 
12.31649  -  10 
log  sin  81°  =    9.99462  -  10 
log  6=    2.32187 
b  =  209.83. 


SOLUTION   OF   OBLIQUE   TRIANGLES  63 

„    ,  c      sin  C 


oiinuany,  10  uuu  c,                      -  = 

a      sin  A 

log  c  =  log  a  +  log  sin  C  —  log  sin 

log  263=    2.41996 

log  sin  47°=    9.86413-10 

12.28409  -  10 

log  sin  81°=    9.99462-10 

logc=    2.28947 

c  =  194.75. 

Check.   -  = ,  from  which 

b      sin  B 

log  c  —  log  b  =  log  sin  C  —  log  sin  B. 

log  c  =  12.28947  -10 

log  sin  C=  19.86413  -20 

log  6=    2.32187 

log  sin  I?  =    9.89653-10 

9.96760  -  10 

9.96760  -  10 

In  all  exercises  in  this  chapter  in  which  the  answers  are  not  given  the  stu- 
dent should  check  his  results.  In  doing  this  he  should  use  the  law  of  sines  and 
also  the  law  of  tangents  (Sec.  46),  which  affords  a  better  check;  when  the 
three  sides  are  given  the  check  A  +  B  +  C=  180°  will  suffice. 


EXERCISES 

Solve  the  following  triangles  : 

1.    Given  A  =  32°  19',  B  =  76°  42',  a  =  268.94,  to  find  C, 
b,  c.  Ans.   C  =  70°  59',  b  =  489.57,  c  =  475.61. 

s   j2.    Given  B  =  51°  47',  C  =  63°  32',  b  =  20.64,  to  find  A,  a,  c. 
Ans.  A  =  64°  41',  a  =  23.748,  c  =  23.517. 

3.  Given  A  =  52°  24'  36",  C  =  74°  12'  16",  a  =  2468.12,  to 
find  B,  b,  c.  Ans.  B  =  53°  23'  8",  b  =  2500.1,  c  =  2997.1. 

4.  Given  A  =  114°  23',  B  =  33°  41  >  =  323,  to  find  C,  b,  c. 

Ans.  b  =  196.68,  c  =  187.58. 
]  5.    Given  A  =  98°  20',  B  =  44°  9',  a  =  964.5,  to  find  C,  b,  c. 
Ans.  b  =  678.98,  c  =  593.64. 
6.    Given  A  =  54°  14',  B  =  44°  27',  a  =  1264,  to  find  C,  5,  c. 
'    Ans.  b  =  1090.9,  c  =  1539.9. 


64  PLANE   TRIGONOMETRY 

7.  Given  B  =  14°  26'  38",    C  =  67°  13'  12",   b  =  23.64,  to 
find  A,  a,  c.  Ans.  a  =  93.776,  c  =  87.384. 

8.  Given  5  =  46°  48',  C  =  84°  23',  b  =  174,  to  find  A,  a,  c. 

Ans.  a  =  179.64,  c  =  237.55. 

9.  Given  A  =  67°  54',  B  =  34°  52',  b  =  435.67,  to  find  C, 
a,  c.  Jm-.  a  =  706.12,  c  =  743.28. 

10.  Given  B  =  31°  12',    C  =  108°  36',   a  =  467,   to  find  vl, 
6,  A,  -4ns.  6  =  374.80,  c  =  685.73. 

11.  Given  A  =  43°  22',  C  =  34°  26',  a  =  346.24,  to  find  £, 
ft,  c. 

12.  Given  B  =  74°  35',  C  =  46°  23',  c  =  246.13,  to  find  A, 
a,  b. 

13.  Given  B  =  52°,  C  =  98°,  c  =  95.46,  to  find  A,  a,  b. 

N  \14.  Given  A  =  23°  18',  B  =  106°,  a  =  34.7,  to  find  C,  b,  c. 

15.  Given  B  =  42°,  C  =  116°,  a  =  564,  to  find  A,  b,  c. 

46.  In  Case  2  we  need  also 


The  Law  of  Tangents.  In  any  triangle  the  sum  of  any  two 
sides  is  to  their  difference  as  the  tangent  of  one  half  the  sum 
of  the  angles  opposite  those  sides  is  to  the  tangent  of  one  half 
their  difference.     Or,  in  symbols, 

a  +  b      tan  \  (A  +  B) 


a  —  b      tan  \  (A  —  B) 

To  prove  this,  we  have,  by  the  law  of  sines, 

a  _  sin  A  # 
~b  ~  sin  B ' 

by  composition  and  division, 

a  +  b  _  sin  A  +  sin  B 
a  —  b      sin  A  —  sin  B 

.:  by  formula  (19),  ±±±  =  '***&+*>, 


(31) 


SOLUTION   OF   OBLIQUE   TRIANGLES  65 

Similarly,  — I—  = :,,„:;>  etc. 

J '  a  —  c       tan  £  (A  —  C) 

47.    Case  2.    Given  two  sides  and  the  angle  included  by  them. 

The  method  of  solution  is  shown  in  the  following  example. 

Given  b  =  360,  c  =  483,  A  =  62°,  to  find  a,  B,  C. 

,     ,  c  +  6      tan{(C  +  5)  A" 

By  the  law  of  tangents,         = — :  • 

J  c-6      tan*(C-#)  -w^ 

From  the  given,  parts,  c/'^rtx's*, 

c  +  6  =  843,  c-b  =  123,  C  +  B  =  180°  -  62°  =  118°  ;  |(C  +  7?)  =  59°.   / 
Substituting  these  values  in  the  above  equation, 
843  _       tan  59° 
123_tani(0-ii)' 
.-.  log  843  -  log  123  =  log  tan  59°  -  log  tan  £  (C  -  £), 
or  log  tan  \  (C  -  B)  =  log  tan  59°  +  log  123  -  log  843. 

log  tan  59°  =  10.22123  -  10 
log  123  =    2.08991 


or 


12.31114  - 

10 

log  843  = 

2.92583 

logtan£(C-B)  = 

9.38531  - 

10 

i(C-B)  = 

13°  38'  57" 

But 

$(C  +  B)  = 

59°. 

Adding, 

C  = 

72°  38"  57" 

Subtracting, 

B  = 

45°  21'    3" 

We  now 

find  a 

by  the  law  of  sines 
a 

c 

sin  A 

sin  G 

log  a 

=  log  c  +  log  sin  A 

—  log  sin  C, 

log  a 

=  log  483  +  log  sin 
log  483  = 

32°  -  log  sin 
:    2.68395 

72°  38'  57 

log  sin  62°  = 

:    9.94593- 

10 

12.62988  - 

10 

log  sin  72°  38' 67"  = 

:    9  97978- 

10 

loga  = 

:    2.65010 

Check. 

a      sin  A                       a- 
b      sin  B 

:  446.79. 

Apply  this  test. 

Q^ 


PLANE    TRIGONOMETRY 


1         EXERCISES 

Solve  the  following  triangles  :  v 

\  » 

1.  'Given  A  =  58°  16',  b  =  656,  c  =  834,  to  find  B,  C,  a.\ 

Ans.  B  =  48°  46'  9",  C  =  72°  57'  51",  a  =  741.85. 

\J  $  2.    Given  A  =  78°  12',  b  =  146,  c  =  174,  to  find  B,  C,  a. 

t.  B  =  44°  45'  17",  C  =  57°  2'  43",  a  =  202.98. 


3.  Given  B  =  53°  18',  a  =  640,  c  =  364,  to  find  A,  C,  b. 
Ans.  A  =  92°  3'  46",  C  =  34°  38'  14",  b  =  513.46. 

4.  Given  C  =  67°  29',  a  =  27,  b  =  22,  to  find  A,  B,  c. 
Ans.  A  =  64°  56'  38",  B  =  47°  34'  22",  c  =  27.532. 

5.  Given  C  =  54°  26',  a  =  123,  &  =  93,  to  find  A,  B,  c. 
Ans.  A  =  77°  53'  46",  B  =  47°  40'  14",  c  =  102.33. 

6.  Given  A  =  56°,  b  =  324,  c  =  456,  to  find  B,  C,  a. 
Ans.  B  =  44°  20'  41",  C  =  79°  39'  19",  a  =  3842.9. 

7.  Given  A  =  56°  10',  b  =  345.6,  c  =  564.8,  to  find  B,  C,  a. 
Ans.  B  =  37°  37'  47",  C  =  86°  12'  13",  a  =  470.18. 

8.  Given  a  =  294,  b  =  114,  C  =  108°  24',  to  find  A,  B,  c. 
Ans.  A  =  53°  27'  1",  B  =  18°  8'  59",  c  =  347.27. 

9.  Given  a  =  32.64,  b  =  46.82,  C  =  44°  12',  to  find  A,  B,  c. 
Ans.  A  =  44°  10'  32",  B  =  91°  37'  28",  c  =  32.654. 

^0.    Given  b  =  3463.8,  a  =  1896.4,  C  =  45°  28',  to  find  A, 
B,  c.  Ans.  A  =  32°  21'  20",  B  =  102°  10'  40",  c  =  2525.9. 

11.  Given  a  =  256,  c  =  438,  B  =  66°  24',  to  find  A,  C,  b. 

12.  Given  b  =  374,  c  =  562,  A  =  104°  32',  to  find  B,  C,  a. 

13.  Given  a  =  235.4,  b  =  392.4,  C  =  53°  18',  to  find  A,  B,  c. 
^J  s/14.  Given  c  =  822,  &  =  654,  A  =  57°  32',  to  find  B,  C,  a. 

15.    Given  A  =  54°  20',  b  =  638,  c  =  362,  to  find  B,  C,  a. 


SOLUTION   OF   OBLIQUE   TRIANGLES 


67 


48.  Case  3.  Given  two  sides  and  an  angle  opposite  one  of 
them. 

Let  the  given  parts  be  a,  b,  A. 

Let  us  first  undertake  to  construct  the  triangle  geometri- 
cally. At  any  point,  as  A  in  the  line  PQ  (Fig.  33),  construct 
an  angle  QAR  equal  to  the  given  angle  A.  On  AR  take  AC 
equal  to  the  given  side  b ;  with  C  as  a  center  and  a  radius  equal 
to  the  other  given  side  a,  describe  an  arc.  Then,  if  the  side 
a  is  less  than  the  perpendicular  CD,  the  arc  will  not  cut  PQ 
and  the  construction  will  be  impossible.  If  the  side  a  is  just 
equal  to  CD,  the  arc  will  touch  PQ  at  D  and  the  right  triangle 


Bt\A         Bh^ 


D 


Bt 


/B3 


Fig.  33 

ADC  fulfills  the  given  conditions.  If  a  is  greater  than  CD 
but  less  than  CA  (b),  the  arc  will  cut  PQ  in  two  points,  Bx  and 
B2,  and  by  joining  each  of  them  to  C  we  form  two  triangles 
ABXC  and  AB2C,  both  of  which  satisfy  the  given  conditions  if 
the  angle  A  be  acute.     Here,  then,  there  are  two  solutions. 

Note  that  the  angles  ABtC  and  AB2C  are  supplementary.    Why  ? 

If  a  is  greater  than  CA,  the  arc  will  cut  PQ  in  two  points, 
but  on  opposite  sides  of  A.  By  joining  B3  with  C  we  form  a 
triangle,  ABZC,  which  fulfills  the  given  conditions ;  but  the 
triangle  AB±C,  formed  by  joining  B±  with  C,  does  not;  for  the 
angle  CAB±  is  not  equal  to  the  given  angle  A.  Hence,  in 
Case  3,  which  is  called  the  ambiguous  case,  the  problem  may 


68  PLANE    TRIGONOMETRY 

be  impossible,  or  it  may  have  one  solution,  or  it  may  have 
two  solutions.  Two  solutions  are  possible  only  when  the  side 
opposite  the  given  angle  is  the  less  of  the  two  given  sides. 

The  student  should  draw  a  separate  figure  for  each  case. 

We   shall   now  illustrate  the   trigonometrical    solution    by 
examples. 
Wl.  Given  A  =  54°,  a  =  64,  b  =  53,  to  find  B,  C,  c. 

_*_■'-.  ,    .  sin  B      sin  A  _      53  x  sin  54° 

By  the  law  of  sines,     — —  = .*.  sin  B  = 

b  a  04 

This  determines  the  value  of  sin  B.  But  any  angle  and  its  supple- 
ment have  the  same  sine.  We  must,  therefore,  accept,  as  values  of  B, 
not  only  the  acute  angle  found  in  the  table  corresponding  to  the  com- 
puted value  of  sin  B,  but  also  its  supplement,  unless  a  reason  can  be 
assigned  for  rejecting  one  of  them.  Applying  logarithms  to  -the  equa- 
tion above,  we  have 

log  sin  B  =  log  53  +  log  sin  54°  —  log  64. 

log  53=    1.72428 
log  sin  54°  =    9.90796  -  10 
11.63224  -  10 
log  64  =    1.80618 
log  sin  B=    9.82606-10 

The  corresponding  angle  in  the  table  is  42°  3'  56".  Its  supplement  is 
137°  56'  4".  This  value  must  be  rejected ;  for,  in  this  example,  b  is  less 
than  a  and,  therefore,  by  geometry,  B  must  be  less  than  A. 

Hence,  we  have  B  =  42°  8'  56". 

To  find  C,  we  have  C  =  180°  -  (A  +  B)  =  83°  56'  4". 

™    ~    ^  ,_  c       sin  C 

To  find  c,  we  have  -  s= • 

a      smA 

log  c  =  log  a  +  log  sin  C  —  log  sin  A. 

log  64=    1.80618 

log  sin  83°  56'  4"  =    9.99756 


11.80374  - 

-10 

log  sin  54°=    9.90796- 

-10 

logc=    1.89578 

Check.   -  =  - — -  • 
6      sin  B 

c  =  78.665. 

Apply  this  test. 

SOLUTION   OF   OBLIQUE    TRIANGLES 


Let  a  =  172,  b  =  213,  A  =  48°. 


69 


To  find  B,  we  have 


B 


sin  A      a 
.-.  log  sin  B  =  log  b  +  log  sin  A  —  log  a 

log  213=    2.32838 
log  sin  48°  =    9.87107  -  10 
12.19945-10 
log  172  ==    2.23553 
log  sin  B  =    9.96392-10 
The  corresponding  angle  in  the  table  is  66°  58" ; 
its  supplement  is  113°  2'.      Both  these  angles 
fulfill  the  condition  B>A,  and  we  have  two 
solutions.      Denoting  the  two  values  of  B  by 
J5i  and  2?2, 

Bx  =  66°  58',         B2  =  113°  2'. 
In  the  triangle  ABXC  (Fig.  34)  denote  the 
angle  A  CBX  by  Ci  and  the  side  ABy  by  cx.    Then, 
d  =  180°  -(A  +  B$  =  65°  2'. 

C\      sin  Oi 
To  find  ci,  we  have 


Fig.  34 


a       sin  A 

.-.  log  Ci  =  log  a  +  log  sin  Ci  - 

-  log  sin  A. 

log  172  =    2.23553 

log  sin  65°  2'  =    9.95739-10 

12.19292  -  10 

log  sin  48°=    9.87107-10 

logCi=    2.32185 

ci  =  209.82. 

Similarly,  in  the  triangle  AB2d  denote  the  angle  ACB^  by  C2  and 

the  side  AB2  by  c2.     Then, 

C2  =  180°  -  (A  +  B2)  =  18°  58' ; 

c-2      sin  C2 
and  to  find  c2,  we  have  —  =  — — -  • 

a       sin  A 

log  sin  A. 


uv 

B                        —  — 

a 

sin  A 

log  c2  = 

log  a  +  log 

sin 

c2 

log  172  = 

2.23553 

log 

sin  18°  58'  = 

9.51191  - 
11.74744  - 

10 
10 

log  sin  48°  = 

9.87107  - 

10 

log  c2  s 

1.87637 

c2  = 

75.227. 

70 


PLANE   TRIGONOMETRY 


Hence,  we  have  for  the  required  parts  the  two  sets  of  values  : 

1.  Bi  =  66°  58',         d  =  65°  2',  cx  =  209.82. 

2.  £2  =  113°2',         C2  =  18°58',         c2  =  75.227. 


EXERCISES 

Solve  the  following  triangles  : 

1.  Given  a  =  540,  b  =  486,  A  =  32°  24',  to  find  B,  C,  c. 

Ans.  B  =  28°  49'  57",  C  =  118°  46'  3",  c  =  883.4. 

2.  Given  a  =  481.34,  b  =  358.26,  A  =  36°  24',  to  find  B, 
C,  c.       Ans.  B  =  26°  12'  38",  C  =  117°  23'  22",  c  =  720.22. 

^\3.    Given  a  =  28,  c  =  15,  A  =  104°,  to  find  B,  C,  b. 

Ans.  B  =  44°  40'  54",  C  =  31°  19'  6",  b  =  20.291. 

1 4.    Given  a  =  22.5,  b  =  40.4,  J  =  42°  14',  to  find  B,  C,  c. 

5.  Given  a  =  88,  b  =  136,  A  =  32°,  to  find  B,  C,  c. 

Ans.  B  =  54°  58'  53",  C  =  93°  V  7",  c  =  165.83. 
Find  the  other  solution. 

6.  Given  b  =  480,  c  =  361,  C  =  46°  4',  to  find  A,  B,  a. 
Ans.  1.  Bx  =  73°  14'  30",  A1  =  60°  41'  30",  ax  =  437.13. 

2.  £2  =  106°  45'  30",  A2  =  27°  10'  30",  a2  =  228.94. 

Given  b  =  356,  c  =  716,  5  =  43°,  to  find  A,  C,  a. 

8.  Given  a  =  342,  b  =  368,  B  =  18°  12'.  to  find  A,  C,  c. 
Ans.  A  =  16°  52'  26",  C  =  144°  55'  34",  c  =  677.05. 

9.  Given  a  =  41.4,  5  =  52.8,  ^  =  40°  19',  to  find  B,  C,  c. 
Ans.  1.  Bx  =  55°  36'  20",    Cx  =  84°  4'  40",    c  =  63.646. 

2.  £2  =  124°  23'  40",  C2  =  15°  17'  20",  c  =  16.872. 

10.  Given  a  =  461,  c  =  513,  C  =  39°  56',  to  find  ^,  j5,  b. 
Ans.  A  =  35°  13'  37",  B  =  104°  50'  23",  b  =  772.56. 

11.  Given  a  =  32.4,  b  =  23.1,  ^  =  36°  17',  to  find  B,  C,  c. 

12.  Given  b  =  186,  c  =  127,  B  =  63°  12',  to  find  A,  C,  a. 


^ 


SOLUTION   OF   OBLIQUE   TRIANGLES 


71 


13.    Given  A  =  42°  14',  a  =  52.5,  b  =  40.4,  to  find  B,  C,  c 
^Ul4.    Given  B  =  46°  28',  b  =  241,  c  =  324,  to  find 


lB,C,c.     I      * 
A,C,a.    ffekj 


15.  Given  C  =  38°  15',  c  =  48,  b  =  36,  to  find  A,  B,  a. 

Without  actually  solving  the  following  triangles,  ascertain 
how  many  solutions  are  possible  in  each  case : 

16.  Given  A  =  36°  40',  a  =  276,  b  =  251. 

17.  Given  B  =  41°  13',  b  =  IS,  a  =  43. 

18.  Given  B  =  43°,  a  =  27,  b  =  36. 

19.  Given  A  =  56°,  a  =  203,  b  =  222. 

20.  Given  B  =  33°  14',  b  =  213,  c  =  165. 


49.  We  shall  now  derive  formulas  for  finding  the  angles  of 
a  triangle  when  the  sides  are  given,  in  order  to  solve  Case  4. 

^Law  of  Cosines.  In  any  triangle  the  square  of  one  side 
equals  the  sum  of  the  squares  of  the  other  two  sides  minus 
twice  the  continued  product  of  those  sides  and  the  cosine  of 
their  included  angle,  or,  in  symbols,  for  the  side  a, 

a2  =  b2  +  c2  -  2  be  cos  A.  (32)  ^ 

To  prove  this,  consider  first  the  case 
where  A   is  acute  (Fig.  35).     Draw  the 
perpendicular  BD.      Then,  DC  =b  —  AD. 
Squaring,  and  adding  BD2  to  each  mem-     A 
ber,  ^G'  s5  . 

DC*  +  BI?  =  b*-2b-AD  +  AD2  +  BD2. 

For  ~DC2  +  BD2  substitute  a2,  and  for  ~AD2  +  BD2  substitute 

c2,  and  we  have 

a2  =  b2-2b-AD  +  c2. 

AD 

=  cos  A. 

c 
.'.  AD  =  c  cos  A. 


But 


i    / 


A? 


_  \ 


\ 


72  PLANE    TRIGONOMETRY 

Substituting  this  value  for  AD, 

a2  =  b2  +  c2-2bc  cos  A. 
If  A  is  obtuse  (Fig.  36),  we  have 
CD  »  b  +  AD. 
Squaring,  and  adding  BD  to  each  member, 

~CD2  +  ~Bff  =  b2  +  2b-AD  +  AD2  +  BD2, 
or  a2  =  b2  +  c2  +  2b-AD. 

AD 

But     =  cos  BAD  —  —  cos  A, 

c 

1j      since  the  cosine  of  an  angle  equals  the 

negative  cosine  of  its  supplement. 

.'.  AD  =  —  c  cos  A. 

Substituting  above,    a2  =  b2  -f  c2  —  2  be  cos  A. 

Similarly,  b2  =  a2  +  c2  —  2  ac  cos  B. 

c2  =  a2  +  62  -  2  a&  cos  C. 

50.    From  the  law  of  cosines  we  can  find  the  cosine  of  any 

angle  of  a  triangle  in  terms  of  the  sides.     Thus,  solving  the 

above  equations,  we  have 

b2  +  c2  —  a2 
cos  A  =      ^ (33) 

cos  B 

cosC 

lab 

These  formulas  can  be  used  to  find  the  angles  when  the 
sides  are  given. 

EXAMPLE 

Given  a  =  8,  b  =  12,  c  =  15,  to  find  A. 

62  +  C2  _  (l2      144  _|_  225  -  64   305   A  0,„0 

cos  A  = =s =  —  =  0.8472. 

2  6c      2  x  12  x  15    360 

.-.  A  =  32°  V  30". 


2  be 

a2 

+  c2- 

b* 

2ac 

a* 

+  b2- 

■c2 

SOLUTION    OF   OBLIQUE   TRIANGLES  73 

51.  The  formulas  of  the  preceding  article  are  not  adapted 
to  logarithmic  computation  [why?],  and  we  proceed  to  derive 
formulas  for  the  functions  of  the  half  angles,  in  which  loga- 
rithms may  be  advantageously  employed. 

To  express  cos  ^  A  in  terms  of  a,  b,  c. 

By  formula  (26), 

2eos2iA  =  1  +  cosA. 

For  cos  A  substitute  its  value  as  given  in  Sec.  50,  and  we 
have 

_2bc  +  b2  +  c2-a2 
2  be 

_(b  +  c)*  -  a2 


2  be 

=  (b  +  c  +  a)(b  +  c  —  a) 
2  be 


Now,  let  b  -\-  c  -\-  a  =  2  s. 

Then,  subtracting  2  a  from  each  member, 
b  -{-  c  —  a  =  2  (s  —  a). 
.   o        21    .       4  *  (s  —  a) 

.'.  2  COSH.4  aa  ^— '-• 

2  be 


=  Js(s-a) 
*       be 


.\cos£A  =  ^-^- L.  (34) 

Since  -J-  A  <  90°,  cos  \  A  is  positive,  and  the  positive  sign  is 
to  be  used  with  the  radical. 


In  like  manner,     cos  -A-  B  =  \— *- 

*       ac 


,     ■  18  (s  —  C) 


74  PLANE   TRIGONOMETRY 

52.    To  express  sin  £  A  in  terms  of  a,  b,  c.     By  formula  (25), 

2  sin2  \A  =1  —  cos  A 

c2-a* 


2  be 

2  be 
-  b*  -c2  + 

a2 

a2- 

2  be 

■  (b2  -2bc-\- 

■c2) 

2  be 
a*-(b-  cy 

2  be 
(a  +  b  —  c)(a  — 

b  +  e) 

2(s- 

2  be 
-c)x2(s- 

■b) 

2  be 

-# 

-b)(s-c) 

-  \ 

be 

<v/<* 

-a)(s-c) 

-\l 

ae 

M 

-a)(s-b) 

.•■mtA=vv    ::    '■  (35) 

Similarly,  sin  £  B 


sin  ., 

*  ab 


53.    To  express  tan  \  A  in  terms  of  a,  b,  c.     From  formulas  (35) 
and  (34),  we  have,  by  division, 


sin  \  A  _     Us  —  b)  (s  —  c)  _^  s  (s  —  a) 
cos  £  A        *  6c  6c 


2 


s  (8  —  a) 


Similarly,  tan  £  B  =  yj(s      "^*      C) 


^  S  (5  -  C) 


SOLUTION   OF   OBLIQUE    TRIANGLES 


75 


54.    We  are  now  prepared  to  solve 

Case  4.  Given  the  three  sides  of  a  triangle,  to  find  the 
three  angles. 

We  may  find  the  values  of  the  half  angles  by  using  the 
formulas  for  the  sine,  the  cosine,  or  the  tangent,  but  a  modifi- 
cation of  the  tangent  formula  furnishes  the  simplest  solution. 

Thus,  from  formula  (36), 

s(s  —  a) 
Multiplying  numerator  and  denominator  by  s  —  a, 

2  s(s  —  a)2 


or 


tan  \A 


J_J(fZ« 

—  n  .    * 


)(,_ft)(<-<.) 


Denoting  ^- ZAr-  -^  by  r,  we  have 


—  a)  (s  —  b)  (s  —  c) 
s 


tan£A  = 

In  like  manner,  tan  \  B  = 


s  —  a 

r 
s-b 

r 

s  —  c 


EXAMPLE 

Given  a  =  126,  b  =  322.6,  c  =  253,  to  find  A,  B, 

a=  126 

b  =  322.6 

c  =  253 
We  have  2s  =  701.6 

s  =  350.8. 
s  -  a  =  224.8. 
s  -  b  =  28.2. 
s-c  =  97.8. 


76  PLANE    TRIGONOMETRY 


log  r  =  £  [log  (s-a)  +  log  (a  -  b)  +  log  (a  -  c)  -  lo$  a]. 

log  (s- a)  =  2. 35180- 

log  (a -6)  =  1.45025 

log(a-c)  =  1.99034 

5.79239 

log  a  =  2.54506 

2)3.24733 

logr  =  1.62367 

Then,  tan|-4  = 


-  a 
log  tan  \A  =  log  r  —  log  (a  —  a). 

logr=  11.62367  -  10 
log(a  -  a)  =    2.35180 
log  tan  M  =    9.27187  -10 
\A  =  10°  35'  33". 
A  =  21°  IT  6". 

T 

tan  i  -B  = 

s  -b 

logr=  1.62367 

log  (a -6)  =  1.45025 

logtani^  =  0.17342 

i£  =  56°8'  49". 

2?  =  112°  17'  38". 

taniC  =  — —  • 
a  —  c 

logr=  11.62367  -10 

log(a  -c)=    1.99034 

log  tan  £  C  =    9.63333  -  10 

i  C  =  23°  15'  39". 

C  =  46°  81'  18". 

Check.   A  +  B+C  =  180°  0'  2". 

How  many  logarithms  had  to  he  looked  up  in  solving  by  this  method  ? 
How  many  would  have  to  be  looked  up  in  finding  the  half  angles  by  the 
sine  formulas  ?    How  many  by  the  cosine  formulas  ? 


EXERCISE 
Determine  A  in  the  above  example  by  the  use  of  formula  (34). 


SOLUTION   OF   OBLIQUE   TRIANGLES  77 

/(ST*- 


C^C^f^T/O 


Solve  the  following  triangles  : 

1.    Given  a  =  10,  ft  =  11,  c  =  13. 

Ans.  A  =  48°  22'  8",  B  =  55°  18'  12",  C  =  76°  19'  42". 
\  J  2.    Given  a  =  342,  *  =  436,  c  =  388. 

Ans.  A  =  48°  36'  56",  B  =  73°  2'  26",  C  =  58°  20'  38". 

3.  Given  a  =  3820,  ft  =  4536,  c  =  2654. 

Ans.  A  =  57°  14'  44",  B  =  87°  0'  4",  C  =  35°  45'  10". 

4.  Given  a  =  4004,  6  =  5056,  c  =  7012. 

4*m.  A  =  34°  7'  20",  B  =  45°  5'  54",  C  =  100°  46'  50". 

5.  Given  a  =  126,  b  =  322.6,  c  =  253. 

Ans.  A  =  21°  11'  6",  5  =  112°  17'  38",  C  =  46°  31'  18". 

6.  Given  a  =  78,  b  =  132,  c  =  146. 

^n*.  -4  =  32°  5'  16",  ,5  =  64°  1'  26",  C  =  83°  53'  22". 

7.  Given  a  =  0.506,  &  =  1.234,  c  =  0.936. 

Ans.  A  =  21°  56'  8",  B  =  114°  21  '1l8",  C  =  43°  42'  34". 

8.  Given  a  =  200,  ft  =  150,  c  =  174. 

-4n*.  A  =  756  50',  £  =  46°  39'  6",  C  =  57°  31'. 

9.  Given  a  =  16.7,  b  =  32.1,  c  =  23. 

^w*.  A  =  29°  51'  50",  B  =  106°  50'  18",  C  =  43°  17'  54". 

10.  Given  a  =  19.61,  b  =  26.41,  0  =  13.54. 

Ans.  A  =  46°  3'  40",  £  =  104°  7'  36",  C  =  29°  48'  50". 

11.  Given  a  =  20.9,  b  =  30.3,  c  =  41.8. 

12.  Given  a  =  521,  ft  =  624,  c  =  728. 

13.  Given  a  =  0.921,  ft  =  1.264,  c  =  0.714. 
XA±.    Given  a  =  1045,  ft  =  1523,  c  =  2286. 

15.    Given  a  =  226,  ft  =  150,  c  =  174. 

55.  To  find  the  area  of  a  triangle.  We  know,  from  geometry, 
that  the  area  of  a  triangle  equals  one  half  the  product  of  its 
base  and  altitude.  That  is,  K  =  \  bp,  where  K  =  area,,  ft  =  base, 
p  =  altitude. 


78  PLANE    TRIGONOMETRY 

1.  To  find  the  area  in  terms  of  two  sides  and  their  included 
angle.  • 

Let  the  given  parts  be  a,  b,  C.     Draw  the  perpendicular 
BD=p  (Figs.  35  and  36).     We  have 

K  =  \bp. 
But  p  =  a  sin  C. 

.\K  =  £absinC.  (38) 

Or,  the  area  of  a  triangle  equals  one  half  the  jwoduct  of  two 
sides  times  the  sine  of  their  included  angle. 

2.  If  one  side  and  two  angles  are  given,  find  another  side 
by  the  law  of  sines  and  apply  formula  (38). 

3.  To  find  the  area  in  terms  of  the  three  sides. 
By  formula  (38),  K  =  \ab  sin  C. 

By  formula  (20),  and  the  formulas  for  sin  %  C  and  cos  \  C 
in  Sees.  52  and  51, 

rt    .                               ^     l(s  —  a)(s  —  b)  Is  (s  -  c) 

sin  C  m  2  sin  £  C  cos  £  C  =  2\* ^ L  X  \      ab 


2     i 

_V*(*-a)(*-&)(s-c). 


Substituting  in  formula  (38), 


K  =  Vs(s-a)(s-b)(s-c).   (39) 

To  find  the  area  of  a  right  triangle,  one  leg  may  be  taken 
as  the  base  and  the  other  as  the  altitude.  If  either  leg  is 
unknown,  find  it  by  using  the  given  parts. 

EXERCISES 
Find  the  area  of  each  of  the  following  triangles : 

1.  a  =  480,  b  =  360,  C  =  62°  42'  16".  Ans.  76778. 

2.  b  =  345,  c  =  423,  A  =  38°  17'. 

3.  A  =  50°  50'  50",  B  =  60°  22'  16",  a  =  481.622. 

Ans.  121197. 
A.    a  =  484,  5  =  346,  c  =  422.  ^ws.  71257. 

5.    a  =  143,  ft  =  257,  c  fc  198.4. 


SOLUTION    OF    OBLIQUE    TRIANGLES 


79 


MISCELLANEOUS   EXERCISES 

J  1.  From  the  ends  of  a  base  line,  EW,  900  feet  long,  observations  were 
taken  on  an  object,  B,  and  the  following  angles  measured:  EWB  = 
85°  19',  WEB  =  67°  51'.      Find  EB  and  WB. 

Ans.  1987.1  feet;  1846.6  feet. 

/A  2.  Two  cyclists  start  at  4  p.m.  from  the  intersection  of  two  straight 
roads  which  meet  at  an  angle  of  40°.  One  travels  at  the  rate  of  8  miles 
per  hour  and  the  other  rides  a  28-inch  wheel,  averaging  120  revolutions 
per  minute.     How  far  apart  are  they  at  5  :  15  p.m.?      Ans.  8.043  miles. 

^3.  In  a  triangle  ABC,  B  =  75°,  C  =  40°,  BC  =  260.  BD  is  drawn  to 
meet  AC  in  D  so  that  angle  ABB  =  35°.     Find  AD.  Ans.  107.4. 

V*4.  An  observer  in  a  balloon  finds  that  the  angles  of  depression  of  two 
points  on  level  ground  and  in  the  same  vertical  plane  with  the  balloon  are 
33°  22'  and  22°  18'  respectively.  If  the  distance  between  the  objects  is 
1200  feet,  find  the  height  of  the  balloon.  Ans.  1304.7  feet. 

5.  From  a  point,  A,  in  a  road  running  north  and  south,  the  bearing 
of  a  church  steeple  was  found  to  be  S.  36°  22'  E. ;  and  from  a  point,  B, 
1986  feet  south  of  A,  it  was  found  to  bear  S.  83°  15'  E.  Find  the  shortest 
distance  from  the  church  to  the  road.  Ans.  1602. 1  feet. 

v  6.  A  roof  is  inclined  to  the  horizontal  an  angle  of  45°,  and  the  ridge- 
pole runs  east  and  west.  At  midday  a  lightning  rod  extending  8  feet 
above  the  ridgepole  casts  a  shadow  on  the  roof  15  feet  long.  Find  the 
altitude  of  the  sun.  Ans.  60°  18'  56". 

7.  From  the  ends  of  a  base  line,  MN,  450  feet  long,  observations  were 
made  on  a  point,  P,  with  the  following  results  :  Z  NMP  =  49°  51', 
/.  MNP  =  110°  50'.     Find  MP  and  NP.  Ans.  1271.4 ;  1039.8. 

8.  A  roof  is  inclined  to  the  horizontal  at  an  angle  of  60°.  A  flagstaff 
standing  on  the  ridgepole  breaks  at  a  point  12  feet  from  its  base,  and  falls 
so  that  the  end  rests  on  the  roof  at  a  point  22  feet  from  the  base  of  the 
staff  and  16  feet  from  the  ridgepole.     Find  the  height  of  the  staff. 

Ans.  43  feet. 

9.  From  two  points,  A'  and  JB',  in  line  with  the  foot  of  a  tower  (J/ the 
more  remote),  the  angles  of  elevation  of  the  top  of  the  tower  are  A  and  B 
respectively.    If  A'B'—  a,  show  that  the  height  of  the  tower  is 

a  sin  A  sin  B 
sin  (B  -  A)  ' 


80  PLANE    TRIGONOMETRY 

10.    A  and  B  are  two  points  in  the  same  horizontal  plane  with  Q,  the 

foot  of  a  tower.     At  A  and  B  the  angles  of  elevation  of  P,  the  top  of  the 

tower,  are  measured;  also  the  angles  QAB,  QBA.     Denoting  these  four 

angles  by  A,  B,  A\  B'  respectively,  and  AB  by  c,  and  the  height  of  the 

t,     l     i,       *u  *  *  C8in  B*      .        .  c  sin  A' 

tower  by  h,  show  that  A  = tan  A  = tan  B. 

sin  (A'  +  BO  sin  (X'  +  B') 

Vll.    In  a  trapezoid  ABCD,  AD  =  BC  =  c,  AB  =  a,  DC  =  b,  DB  =  d, 

Z^BD=m,  Z.AZXB  =  w.     Show  that  d  =  b  sm  (m  +  n) . 

sin  (2  m  +  n) 

v  .12.    If  an  angle  of  a  triangle  be  doubled  while  the  including  sides  remain 
constant,  will  the  area  be  increased  or  diminished  ? 


S+. 


A  base  line,  EW,  900  feet  long  was  measured  on  the  top  of  a  dam 
and  the  following  angles  were  read  :  WEA  =  110°  50',  BEW=  67°  51', 
AWE=z  49°  51',  EWB  =  85°  19'.     Find  ^1B.  Arts.  1492.4  feet. 

14.  The  angles  of  elevation  of  the  top  of  a  tower  from  two  points,  A 
and  jB,  in  line  with  its  base  are  45°  and  30°  respectively.    If  AB  =  a,  show 

that  the  height  of  the  tower  is  — ==:  • 

V4-2V3 

15.  A  boy  flies  a  kite  over  a  level  plane  when  the  sun  is  due  south ; 
the  altitude  of  the  sun  is  60°  and  the  wind  blows  in  the  direction  N.  50°  W. 
The  line  from  the  boy's  position  to  the  shadow  of  the  kite  is  430  feet  long 
and  bears  N.  32°  W.     Find  the  height  of  the  kite.       Ans.  300.44  feet. 


<V 


/  D  ^ 


1 


.PTER   VII 


GEOMETRICAL   APPLICATIONS 


56.  To  find  the  radius  R  of  the  circumcircle.  A  circle  passing 
through,  the  vertices  of  a  triangle  is  called  the  circumscribed 
circle,  or  circumcircle  of  the  triangle.  Its  center  is  called  the 
circumcenter.  Let  0  (Fig.  &7)  be  the  circumcenter  of  the 
Draw  the  perpendicular  OD.     Then, 

Z.BOD  =  a\  Why? 
BD\  ja 
~R\     R  ' 


triangle  ABC. 


' .  sin  A  = 


R  = 


Similarly,   R  =  ^—^  =  £^ 

-r-r  a  b 

Hence,         R  =  7.— — -  =  -^—. — -  =t=  77-^ — 7; 
2smA      2smB  \  2  sin  (7 


Fig.  37 


57.    To  find  the  area  of  a  triangle  in  terms  of  the  radius  of  the 
circumcircle.     By  formula  (38),  we  have 

K  =  \ab  sin  C. 
c 
2R' 
abc 

Ir' 


But 


sin  C 


K 


EXERCISE 

Prove  K  =  2  R2  sin  A  sin  B  sin  C. 

81 


\ 


82 


PLANE    TRIGONOMETRY 


or 


58.  To  find  the  radius  of   the  incircle  of   a  triangle.      The 

inscribed  circle  of  a  triangle  is  called  the  incircle.  Let  O  be 
the  center  of  the  incircle  of  the  triangle  ABC  (Fig.  38)  and 
r  its  radius.  Then,  by  geometry,  0  is  the  intersection  of  the 
bisectors  of  the  angles  A,  B,  C.     Draw  OD,  OE,  OF  to  the 

points  of  contact.     Then, 

BD  +  DC  =  a. 

CE  +  EA=  b. 

AF+  FB  =  c. 

Adding,   and   remembering  that, 

by  geometry,  AF  =  AE,  DC  =  CE, 

FB  =  DB, 

2AF+2BD  +  2CD  =  a  +  b  +  c  =  2s. 

.'.AF  +  (BD+  CD)  =  s, 

AF  +  a  =  s. 

.'.  AF=s  —  a. 

OF 

AF' 

r 
tan  \  A  — > 

s  —  a 

r  =  (s  —  a)  tan  \  A. 
Similarly,  r  =  (s  —  b)  tan  ^  B. 

r  =  (s  —  c)  tan  ^  C. 
Hence,    r  =  (s  —  a)  tan  \A  ={s  —  V)  tan  J  £  =  (s  —  c)  tan  \  C. 

59.  To  find  the  radii  of  the  escribed  circles.  A  circle  touching 
one  side  of  a  triangle  and  the  other  two  produced  is  called  an 
escribed  circle  of  the  triangle.  Let  0  (Fig.  39)  be  the  center 
of  the  escribed  circle  touching  the  side  BC  (a)  of  the  triangle 
ABC.  Denote  its  radius  by  ra.  The  points  of  contact  being 
D,  E,  F,  we  have,  by  geometry, 

AE  =  AF,     BD  =  BF%      CD  =  CE. 


Then, 


tan  OAF 


or 


or 


GEOMETRICAL   APPLICATIONS 


83 


But 


.'  .2  s  =  AB  +  BD  +  DC  +  AC 
=  AB  +  BF+CE  +  AC 
=AF+AE 
=  2AF. 

'.AF  =  s. 

OAF  =  \A.    Why? 


".  tan 


v 
A  =  -±- 
s 


Similarly, 


.-.  ra  —  stan  \A. 
rb  =  s  tan  £  B. 
rc  =  s  tan  £  C. 


60.    To  find  the  area  of  a  triangle  in  terms  of  the  radius  of  the 
incircle.     We  have,  from  Fig.  38, 

ABOC  =\ra. 


AAOC  =  \rb. 
AAOB  =  \rc. 


By  addition, 


or 


AABC  =  ±r(a  +  b  +  c), 


61.    To  express  K  in  terms  of  the  radius  of  the  incircle  and  the 
radii  of  the  escribed  circles.     From  Sec.  54, 

tan  \  A  = 


From  Sec.  59, 


But 


—  a 

tan  \  A  =  ^  • 

2  s 

•       r      =  V 

'  s  —  a       s 

.- .  rs  =  ra(s  —  a). 

K  =  rs. 
.'.K  =  ra(s-a). 


84 


PLANE   TRIGONOMETRY 


Similarly,  K  =  rb(s  —  b). 

K  =  rc(s-c). 
By  multiplication,        K 4  =  rrarbrcs  (s  —  a)(s  —  b)  (s  —  c). 
But,  by  formula  (39), 

s(s  —  a)(s  —  b)  (s  —  c)  =  K2. 

•'•  K*  =  rrarhrci 
or  if  =  ^/rrarbrc. 

62.    We  have,  from  equations  in  the  preceding  section, 
1  _s  —  a 

1  _  s  —  c 

,  .....         Ill       3<~(a+-Hc)        s        s       1 

.'.  by  addition, 1 1 —  = » -  =—=—=- 

ra       rb       rc  A'  K       rs       r 


EXERCISE 
Prove  the  relation  —  +  —  +  —  =  -,  where  ha,  hb,  hc  are. 

"'a  "'b  "'c  T 

the  altitudes  of  the  triangle  ABC  and  r  the  radius  of  the 

incircle. 

63.  To  find  the  distance  from  cir- 
cumcenter  to  the  incenter  in  terms  of 
R  and  r.  Let  0  be  the  circum center 
and  /  the  incenter  of  the  triangle 
ABC  (Fig.  40).  Draw  the  diameter 
,G  DE  perpendicular  to  BC.  I  is  in 
the  bisector  of  the  angle  BAC. 
.'.  A I  produced  passes  through  D. 
Through  /  draw  the  diameter  HIG. 
Then,  by  geometry, 


GEOMETRICAL    APPLICATIONS  85 

HIx  IG  =  AIx  ID, 

or  (R  +  01)  (R  -  01)  =  AI  x  ID. 

.'.R2-  012  =  AIx  ID. 

To  find  the  value  of  01,  therefore,  it  is  necessary  to  find 
values  for  A I  and  ID.     Draw  IF  perpendicular  to  AC.    Then 

IF  =  r, 
and,  from  the  right  triangle  AFI, 

r 
sm  \A 

Now,  prove  DI  =  DC.     Z  DIC  is  an  exterior  angle  of  the 
triangle  AIC. 

.-.  Z  D/C  =  ZIAC  +  ZACI=IA  +%C. 
From  the  figure,        Z  DCI  =  Z  £>C£  +  Z  £C/ 

=  £yl+£C.      Why? 
.-.  ZDIC  =  ZDCI. 
.'.  ID  =  DC. 
Now,  from  the  right  triangle  DCE, 

DC  =  DE  sin  ZXEC. 
But  DE  =  2R, 

and  ZMC  =  }i 

.\DC  =  /Z):=2i2sin£,4. 
Substituting  these  values  for  A I  and  ID  above, 

R*-  oJ2=    .   T    A  X2i?sin^. 
sm  £  A  2 

.'.  0/  =  ViT2  -  2  iZr. 

64.    To  find  the  area  of  an  inscribed  quadrilateral  in  terms  of  its 
sides.     In  Fig.  41  let 

AB  =  a,  BC  =  b,   CD  =  c,  DA  =  d,  area  of  ABCD  =  S. 
Now,  area  of  A  ABC  =  ^  ab  sin  B, 


86 


PLANE    TRIGONOMETRY 


and      area  of  A  ABC  —  \cd  sin  B  =\cd  sin  B, 

since  Z  B  is  the  supplement  of  Z  B. 
Why  so  ? 

.'.<$=} (a&  +  cd) sin  5. 
It  is  necessary,  therefore,  to  find  sin  B. 
In  triangle  ABC  we  have,  by  the  law 
of  cosines  (Sec.  49), 

Zc2  =  a2  +  b2-2ab  cos  5, 
and  in  triangle  ABC, 

AC2  =  c2  +  d2-2  cd  cos  B. 
Equate  the  two  values  of  A  C  ,  and  for  cos  B  substitute 
—  cos  B,  and  solve  for  cos  B.     We  find 

a2  +  b2  _  c2  _  d2 


cos  B  = 


sin2£  =  l  -cos2£ 


-[*- 


2  (a£  +  erf) 

2  +  £2  _  C2  _  ^2 


:] 


2(a6'+cd) 
_  4  (afl  +  cd)2  -  (a2  +  b2-c2-  d2)2 

4(ab  +  cd)2 
_[2(ab+cd)  +  (a2+b2-c2-d2)l[2(ab+cd)-(a2+b2-c2-d2)] 
~  4(ab+cd)2 

_  [a2+2ab+b2-(c2-2cd+d2)][c2+2cd+d2-(a2-2ab+b2)] 

4(ab+cd)2 
^[(a  +  b)2-(c-d)2][(c  +  d)2-(a-b)2] 

±(ab  +  cd)2 
_  (a+b+c—d)(a  +  b  —  c  +  d)(c-\-d  +  a  —  b)(c  +  d  —  a  +  b) 
±(ab  +  cd)2 

Let  2s  =  a  +  b  +  c  +  d. 

Then,  2  s  —  2  a  =  b  +  c  +  d  —  a  =  2  (s  —  a). 

2s-2b=a-b  +  c  +  d  =  2(s-b). 

2s  —  2c  =a  +  b-c  +  d  =  2{s  —  c). 

2s  —  2d  =  a  +  b  +  c  —  d  =  2(s  —  d). 


GEOMETRICAL   APPLICATIONS  87 

«  (ab  +  cd)2 


Substituting  this  value  of  sin  B  in  the  expression  for  S 
above,  we  have 

S  =  %(ab  +  cd)  X  ^-p-^  ^(s-a)(s-b)(s-c)(s-d), 

or  S  =  V(s  —  a)  (s  —  b)  (s  —  c)  (s  —  d). 

65.  Angles  that  can  be  constructed  geometrically,  and  their 
Functions.  Let  us  inquire  what  angles,  expressible  exactly  in 
degrees,  i.e.,  containing  an  integral  number  of  degrees,  can  be 
constructed  with  ruler  and  compass. 

The  construction  can  be  made  to  depend  in  each  case  upon 
the  inscription  of  a  regular  polygon  in  a  circle.  We  can,  by 
geometry,   inscribe   regular   polygons   of  _ 

three,  four,  five,  fifteen  sides.  The  cen-  ^ 
tral  angles  subtended  by  the  sides  of 
these  polygons  are  120°,  90°,  72°,  24° 
respectively.  The  regular  polygon  of 
seventeen  sides  and  certain  others  may 
also  be  inscribed  with  ruler  and  compass,  but  the  correspond- 
ing central  angles  are  not  expressible  exactly  in  degrees,  and 
these  cases  will  not  be  discussed. 

Since  an  angle  may  be  bisected,  we  may  construct  the  follow- 
ing angles,  without  introducing  fractions  of  a  degree : 

120°,  60°,  30°,  15° ;  90°,  45° ;  72°,  36°,  18°,  9° ;  24°,  12°,  6°,  3°. 

Each  of  these  angles  is  a  multiple  of  3°.  Corresponding 
to  each  of  these  central  angles,  we  may  construct  a  regular 
inscribed  polygon,  and  the  interior  angles  of  the  polygons 
must  be  added  to  the  list  of  angles  that  can  be  constructed 
with  ruler  and  compass.  But  if  ABD  (Fig.  42)  be  one  of  such 
interior  angles,  we  have 

OBA  =  J  ABD  =  90°  -  \  O.     .'.  ABD  =  180°  -  0. 


G      B 

Fig.  42 


88  PLANE    TRIGONOMETRY 

And  since  180°  and  0  are  multiples  of  3°,  ABB  is  a  multiple 
of  3° ;  so  also  are  the  angles  formed  by  bisecting  these  interior 
angles  successively  as  many  times  as  it  can  be  done  without 
introducing  fractions  of  a  degree.  Hence,  every  angle  express- 
ible exactly  in  degrees,  that  can  be  constructed  by  ruler  and 
compass,  4s  a  multiple  of  3°.  All  such  angles  are  included  in 
the  series  3°,  6°,  9°,  12°,  15°, 

The  angle  3°  is  constructed,  as  we  have  seen,  by  beginning 
the  central  angle  24°  and  bisecting  successively  three  times, 
giving  the  angles  12°,  6°,  3°. 

To  construct  any  multiple  of  3°  we  have  only  to  apply  the 
chord  of  this  angle  the  requisite  number  of  times  to  the  cir- 
cumference, and  join  the  first  and  last  points  of  division  to 
the  center. 

If  we  admit  now  angles  not  expressible  exactly  in  degrees, 
we  may,  by  successive  bisections,  construct  any  angle  included 

N 
in  the  form  —  t  where  N  is  3°,  or  any  multiple  of  3°,  and  n 
Jt 

any  integer.     And  since  angles  may  be  added  or  subtracted, 
we  may  also  construct  any  angle  included  in  the  more  general 

N      M 
form  —  ±  —  9  where  N  and  M  are  multiples  of  3°,  and  n  and 

m  any  integers. 

EXERCISES 

1.  Make  a  list  of  all  the  acute  angles  expressible  in  degrees 
that  can  be  constructed  geometrically. 

2.  Assume  values  for  N  and  n  in  the  expression  —  and 

determine  four  angles,  each  less  than  3°,  that  can  be  con- 
structed geometrically.     Thus,  let 

N=6°,n  =  3.     Then,  |^  |  =  45'. 

We  shall  now  show  how  to  find  the  functions  of  3°,  and 
from    them    the    functions    of    any   multiple   of   3°.      Since 


GEOMETRICAL    APPLICATIONS  89 

3°  =  18°  —  15°,  we  shall  first  show  how  to  find  the  functions 

of  15°  and  18°. 

By  formula  (25), 

V3 
2  sin215°  =  1  -  cos  30°  =  1  -  -^, 

and,  by  formula  (26), 

2  cos215°  =  1  +  cos  30°  =  1  +  —-* 

.    .    1KO     V2~~Vl 

.'.  sin  15  = = 


iKo       V2  +  V3 
cos  15   = - 


From  either  of  these  the  other  functions  may  be  found. 
For  the  angle  18°  let  AB  —  x  be  the  side  of  the  regular 
inscribed  decagon,  and  let  AO  =  R  (Fig.  42). 
Then,  by  geometry, 

R  —  x  :  x  =  x  :  R  ; 

.                                                 R(V5-1) 
whence,  x  =  — s — '-  • 

.     •    1QO      AC       V5-1 
••Sml8   =AO  =  — 4— 


.  .  COS  1<5     =  — 

2V2 

Now, 

sin  3°  =  sin  (18°  -  15°)  =  sin  18°  cos  15°  -  cos  18°  sin  15°, 

00      V5-1    V2  +  V3      VsTVi   V2- V3 

or    sin  3   = = — 

4  2  2V2  2 

The  other  functions  of  3°  may  be  found  in  a  similar  way, 
or  by  Sec.  32. 

For  sin  6°,  put  x  =  30°  in  formula  (20). 
For  sin  9°,  put  x  =  6°,  y  =  3°  in  formula  (9). 


90  PLANE   TRIGONOMETRY 

By  successive  applications  of  formulas  (9)  or  (11)  we  can  find 
the  sine  of  the  sum  or  difference  of  any  two  multiples  of  3° 
whose  sines  and  cosines  have  been  previously  found.  Having 
found  the  sine  and  cosine  of  any  angle  in  the  series  of  multiples 
of  3°,  we  may  find  the  sine  and  cosine  of  half  the  angle  by 
formulas  (25)  and  (26).  We  may,  therefore,  find  the  sine,  the 
cosine,  and  the  other  functions  of  any  angle  included  in  the 

.         N.M       .  , 

form  —  ±  —  >  given  above. 

MISCELLANEOUS   EXERCISES 

1.  Find  the  values  of  the  following  in  the  form  of  surds  as  above  : 
cos  3°,  cos  33°,  sin  57°,  cos  6°,  sin  93°,  cos  12°,  sin  36°,  cos  7°  30'. 

2.  Prove  the  relation  P2  =  H2  +  D2,  where  P,  H,  D  are  the  sides  of 
the  regular  inscribed  pentagon,  hexagon,  decagon  respectively. 

D  has  been  determined  above ;  from  a  figure  find  P  =  2  r  sin  36°. 

3.  Find  the  length  of  the  side  of  the  regular  inscribed  polygon  of  60 
sides  in  terms  of  the  radius  B. 


CHAPTER   VIII 


RADIAN  MEASURE,   OR   CIRCULAR  MEASURE;    INVERSE 
FUNCTIONS;    GENERAL  EXPRESSIONS 

66.  Radian  Measure,  or  Circular  Measure.  To  measure  any 
magnitude  is  to  find  its  ratio  to  some  magnitude  of  the  same 
kind  taken  as  a  unit.  So,  to  measure  an  angle  is  to  find  its 
ratio  to  some  constant  angle  taken  as  a 
unit.  What  the  unit  angle  shall  be,  is 
entirely  arbitrary.  A  unit  in  common 
use  is  an  angle  of  one  degree;  other 
units  are  angles  of  one  minute,  one 
second,  a  right  angle.  Still  another 
unit,  whose  use  is  convenient  in  ana- 
lytic work,  is  an  angle  called  a  radian. 
Let  A  (Fig.  43)  be  the  center  of  a 
circle,  and  BA  C  a  central  angle  whose  sides  intercept  an  arc 
BC  equal  in  length  to  the  radius  r.  The  angle  so  determined 
is  called  a  radian,  or  the  unit  of  circular  measure.  The  cir- 
cular measure  of  an  angle,  therefore,  is  its  ratio  to  one  radian. 
To  obtain  a  relation  between  degrees  and  radians,  compare  the 
angle  BAC  with  the  straight  angle  BAD.  By  geometry,  cen- 
tral angles  in  the  same  circle  have  the  same  ratio  as  their  arcs. 

/.BAD      arc  BCD 


Fig.  43 


But 

and,  by  hypothesis, 

Hence, 


'  Z.BAC~   arc  BC 
arc  BCD  =z  irr  [why?], 
arc  BC 


one  straight  angle 
one  radian 


?*. 


irr 


91 


\ 


D2  PLANE   TRIGONOMETRY 

.".  one  straight  angle  equals  it  times  one  radian, 

or  180  degrees  =  it  radians.  (40) 

This  relation  shows  that  the  radian  is  a  constant  angle  and 
enables  us  to  convert  degrees  into  radians,  or  radians  into 
degrees.     By  dividing  each  member  of  formula  (40)  by  7r,  we 

180° 
find  one  radian  to  be  =  57°.29577  +  or  57°.3  nearly. 

EXAMPLES 

1.  How  many  radians  in  an  angle  of  105°  ?    of  90°  ? 

From  formula  (40),  1°  = radians. 

v     "  180 

.-.  105°  ss radians  ==  T%  it  radians. 

180  T- 

For  it  substitute  3.1416,  and  we  find 

105°  =  1.8326  radians. 

«.     .,     ,  ™r>       90  7T        ,.  it 

Similarly,  90°  = radians  =  -  radians. 

J'  180  2 

The  circular  measure  of  a  right  angle  is,  therefore,  —  ;  and  for  a  straight 
angle  it  is  re. 

2.  Express  2\  radians  in  degrees. 

180 
From  formula  (40),       1  radian  = degrees. 

.-.  2^  radians  =  -  x degrees. 

L  It 

For  it  substitute  3.1416,  and  reduce. 

The  symbols  °,  ',  "  for  degrees,  minutes,  seconds  are  familiar.  When 
the  numerical  value  of  an  angle  is  given  without  the  use  of  such  symbols 
or  without  naming  the  unit,  it  is  understood  that  the  unit  is  one  radian. 

2  2  it 

Thus,  an  angle  -  means  -  of  a  radian;  an  angle  —  means  one  half  it  radians. 

3  3  2 

As  a  symbol  for  the  unit  angle,  one  radian,  it  is  proposed  to  use  the 
Greek  letter  p,  writing  2p,  f  p,  f  itp,  for  two  radians,  three  fourths  of  a 
radian,  three  fifths  of  it  radians. 


RADIAN   MEASURE,  OR   CIRCULAR   MEASURE       93 

J 67.  To  express  the  length  of  an  arc  in  terms  of  its  central  angle 
and  the  radius  of  the  circle.  Let  s  be  the  number  of  linear  units 
in  the  arc  BCP  (Fig.  43)  ;  0  the  number  of  radians  in  its  cen- 
tral angle  BAP.     Then,  by  geometry, 

arc  BCP      Z  BAP 


arc  BC 

ABAC 

or 

s 
r 

e 

since 

ABAC 

is 

one 

radian. 

.".  s 

=  r6. 

(41) 


EXERCISES 


1.  Express  in  radians  360°,  225°,  120°,  90°,  60°,  45°,  30°,  18°. 

Give  the  results  first  in  terms  of  ir;  then  substitute  the  numerical  value 
of  r.     Thus,  360°=  2  tt  radians  =  6.2832 p. 

2.  Express  in  degrees  one  radian,  2.4  o,  ±  7rp,  T5^  7rp,  1.6  p. 
Reduce  decimals  of  a  degree  to  minutes  and  seconds. 

3.  Express  in  radians  42°.25,  64°  15'  24",  144°  18'  36", 
5°  44'. 

4.  Find  in  degrees  and  also  in  radians  the  central  angles 
subtended  by  the  sides  of  regular  polygons  of  3,  7,  8,  9,  10, 
15,  n  sides. 

5.  Find  the  lengths  of  the  subtended  arcs  in  Example  4, 
if  the  radius  of  the  circle  is  16  feet. 

6.  In  a  circle  whose  radius  is  236  feet,  what  is  the  length 
of  the  arc  corresponding  to  a  central  angle  of  40°?  of  128°  43'? 

7.  In  a  circle  whose  radius  is  16  inches,  the  length  of  an 
arc  is  3  feet.  How  many  radians  in  its  central  angle  ?  How 
many  degrees  ? 

8.  Express  in  radians  the  angles  of  the  following  regular 
polygons :  a  square,  a  pentagon,  a  hexagon,  an  octagon,  a 
decagon,  a  dodecagon. 


\_ 


V 


1^ 


94  PLANE    TRIGONOMETRY 

9.    Regarding  the  earth  as  a  sphere,  radius  3956  miles,  what 
is  the  length  of  an  arc  of  1°  ?   of  1"  ? 

10.  Prove  that  the  area  of  a  sector  of  a  circle  is  \  r20,  where 
6  is  the  number  of  radians  in  the  central  angle  and  r  the 
radius. 

11.  Find  the  number  of  seconds  in  one  radian. 

*68.    The  57.3  Rule.     Let  ABC  (Fig.  44)  be  a  triangle  in 

which  the  angle  BAC  is  small  and  the  angle  B  nearly  90°. 

Then  the  including  sides  will  be  nearly  equal,  and  the  short 

side  BC,  or  a,  will  differ  in  length  but 
little  from  the  arc  BD,  whose  center 
is  at  A  and  whose  radius  is  c. 
D  G       Let  A   be  the  number  of  degrees 
and  0  the  number  of  radians  in  the 

angle  BA  C.     Then,  since  (Sec.  66)  one  radian  equals  57.3°, 

we  have 

c  •  A 
Also,  by  formula  (41),     BD  =  c  •  0  ■=  --—^  • 

Replacing  BD  by  a,  which  is  nearly  equal  to  it,  and  clearing 
of  fractions,  we  have  approximately 

a  X  57.3  =  c  X  A.  (42) 

This  relation  is  known  as  the  57.3  Rule.  It  may  be  stated 
as  follows : 

The  short  side  multiplied  by  57.3  equals  one  of  the  long  sides 
multiplied  by  the  small  angle  in  degrees. 

Formula  (42)  may  be  used  to  find  the  short  side  BC  when 
A  is  given,  or  A  when  BC  is  given.  This  method  is  suffi- 
ciently accurate  for  practical  purposes  when  the  small  angle 
is  less  than  6°. 


RADIAN  MEASURE,  OR  CIRCULAR  MEASURE        95 

EXERCISES 

Apply  the  57.3  Rule  to  the  solution  of  the  following 
problems : 

1.  A  straight  roadway  is  1320  feet  long  and  has  a  rise  of 
21  feet.    Find  its  inclination  to  the  horizontal.    Ans.  54'  42". 

2.  The  diameter  of  the  moon  subtends  an  angle  of  31'  at 
the  earth,  and  the  distance  from  the  moon  to  the  earth  is 
about  240,000  miles.     Find  the  diameter  of  the  moon. 

Ans.  2164  miles. 

3.  The  diameter  of  the  sun  subtends  an  angle  of  32'  at 
the  earth,  and  the  distance  from  £he  sun  to  the  earth  is  about 
93,000,000  miles.     Find  the  diameter  of  the  sun. 

Ans.  865,(p0  miles. 

4.  The  diameter  of  the  earth,  7912  miles,  subtends  an  angle 
of  17".6  at  the  sun.  Find  the  distance  from  the  earth  to  the 
sun.  Ans.  92,732,236  miles. 

5.  The  radius  of  the  earth's  orbit  is  about  93,000,000 
miles.  If  the  angle  formed  by  lines  drawn  from  the  nearest 
fixed  star  to  the  extremities  of  this  radius  form  an  angle  of 
0".913,  what  is  the  distance  from  the  star  to  the  earth  and 
how  long  does  it  take  a  ray  of  light  to  traverse  this  distance, 
the  velocity  of  light  being  about  186,000  miles  per  second  ? 

6.  A  surveyor,  wishing  to  know  the  distance  between  two 
points,  A  and  B,  on  opposite  sides  of  a  river,  measured  the  angle 
BAC  —  5°  44';  the  distance  BC  was  found  by  an  assistant  to 
be  326  feet.  BC  being  nearly  perpendicular  to  AB,  find  the 
length  of  AB.  Ans.  3258  feet. 

7.  Telescopes  at  the  ends  of  a  line  200  feet  long  on  the 
deck  of  a  ship  are  turned  upon  a  point  on  a  distant  fort. 
The  lines  of  sight  are  found  to  make  angles  of  89°  17'  and 
89°  13'  respectively  with  the  given  line.  Find  the  distance 
from  the  ship  to  the  fort.  Ans.  7640  feet. 


96  PLANE    TRIGONOMETRY 

\ 

8.  What  is  the  angle  subtended  by  the  radius  of  the 
earth's  orbit  at  the  pole  star,  if  it  takes  its  light  42.4  years 
to  reach  the  earth  ? 

9.  The  diameter  of  a  sphere  is  1  yard.  At  what  distance 
will  it  subtend  an  angle  of  1"  ?  ' 

10.  Find  the  length  of  an  arc  of  1'  on  the  earth's  surface, 
regarding  the  earth  as  a  sphere,  radius  3956  miles. 

INVERSE   FUNCTIONS,    OR   ANTIFUNCTIONS 

69.    From  the  right  triangle  A  CB  (Fig.  45),  we  have 

sin  A  =  £ . 
We  write       A  =  sin-1 §, 
which  is  read :   A  is  an  angle  whose 
sine  is  f . 

The  —  1  is  not  used  here  as  an  expo- 
nent.    From  the  figure,  A  is  also  an 
FlG  ^  angle   whose    cosine   is    f.      This    is 

expressed,  similarly,  A  =  cos_1£;  so 
for   the   other   functions. 

Likewise,  sin-1  a;  is  an  abbreviation  for  the  angle  whose 
sine  is  x.  sin_1x  is  also  called  the  antisine  of  x,  or  the 
inverse  sine  of  x. 

A  ss  sin-1  a?  and  x  —  sin  A  are  equivalent  statements.  We 
have  seen  (Sec.  25)  that  one  function  does  not  determine  an 
angle  absolutely.  For  example,  any  angle  and  its  supplement 
have  the  same  sine;  and  if  either  of  them  be  increased  by 
360°,  the  resulting  angles  will  also  have  the  same  sine.  The 
least  positive  value  of  an  antifunction  is  called  its  principal 
value.  Thus,  the  least  positive  angle  having  £  for  its  sine  is 
30°;  i.e.,  the  principal  value  of  sin_1i  is  30°;  the  principal 
value  of  sec_1(—  2)  is  120°.  The  principal  value  of  an  anti- 
function  is  sometimes  indicated  by  the  use  of  a  capital  letter. 
Thus,  Sin-1i  =  30°. 


INVERSE   FUNCTIONS,   OR   ANTIFUNCTIONS        97 

EXERCISES 

1.  Construct  the  following  angles  : 

Cos-H,     Tan-1!,     Csc-1(-2),     Tan"1^. 

2.  Find  the  remaining  functions  of  each  of  the  above  angles. 

3.  What  is  the  tangent  of  Sin"1  j§  ? 

4.  Express  tan-1 6  as  an  anticosine. 

Solution.     Let                        A  =  tan-1 6. 
Then,  tan  A  =  b. 

.-.  sec  A  bs  Vl  +  b2, 
1 


and  cos  A 


Vl  +  62 
A  =  tan— 1  b  =  cos-  * 


Vl  +62 


7T 

5.  Show  that  Sin-1  a  -f-  Cos-1  a-  =  —  p,  a  being  numerically 
not  greater  than  1. 

6.  Express  in  radians  Cot-1 1,  Sec-1  — js.  t  sin-1 1,  cos-1 0. 

r  —  y  y 

7.  Prove  cos-1 =  versin-1  -  • 

^^_       r  r 

A-    &-$.    Prove  tan- xa  + tan- ^  =  tan-1  q^t-r- 

I  —  ao 

Write  x  =  tan- 1  a,  y  =  tan-  *  6. 

Then,  tan  x  =  a,  tan  y  =  b. 

Find  tan  (x  +  y)  in  terms  of  a  and  b  by  formula  (13). 

Prove  the  following  relations  : 
9.    Tan-1(-l)+Tan-1|  =  Tan-1(-|). 

10.  Tan- 1 1  +  2  Tan"1  J  =  jp. 

11.  Bfo-*i  +  GBtrl$  =  jP.  x^vu    ty( 

12.  4Tan-1i-Tan-1^  =  jP. 

13.  Tan"1  J  -  Tan"1  TV  +  Tan"1  H  =  jp. 


98 


PLANE   TRIGONOMETRY 


GENERAL   VALUES   OF  ANTIFUNCTIONS 


*  70.   To  find  an  expression  which  shall  include  all  angles  having 

a  given  sine.     Let  the  value  of  the  given  sine  be  s.     s  must,  of 

course,  be  less  than  1,  and  we  shall  suppose  it  to  be  positive. 

Let  XOP  (Fig.  46)  be  the  least  positive  angle  that  has  its 

sine  equal  to  s.     Let  the  value  of  XOP  in  radians  be  0. 

Its  supplement,  XOP',  or  ir  —  0,  will  also  have  s  for  its  sine ; 
and  if  0  be  increased  by  2  it  radians,  or  m  times  2  it  radians 
where  m  is  any  whole  number,  the  terminal  line  will  again 
come  into  the  position  OP  and  each  of  the  angles  formed  will 
have  its  sine  equal  to.s.     All  such  angles  are  included  in  the 

expression 

m  •  2  7r  -h  6,  or  2  m7r  +  6.     (a) 

Likewise,  if  XOP',  or  it  —  0,  be 
increased  by  m  times  2  it,  each  of 
the  resulting  angles  will  have  s  for 
its  sine.  All  such  angles  are  in- 
cluded in  the  expression 

m  ■  2  ir  +  (tt  —  6), 
or  (2m  +  l)7r-0.  (b) 

And,  conversely,  if  the  sine  of 
an  angle  be  s,  its  terminal  line  must  occupy  one  of  the  posi- 
tions OP  or  OP'. 

Comparing  these  two  expressions,  we  see  that  when  the  coeffi- 
cient of  7r  is  even,  as  in  (a)  (2  m  being  necessarily  even),  the 
sign  of  6  is  + ;  when  the  coefficient  of  it  is  odd,  as  in  (b) 
(2m  +  l  being  necessarily  odd),  the  sign  of  0  is  — .  There- 
fore the  angles  6,  2  mir  +  6,  (2  m  +  1)  rr  —  6,  and  no  others,  all 
have  the  given  sine  s.     They  may  all  be  included  in  the  general 


expression  nir  +  (■ 
number.     For,  if  i 


l)n0,  where  n  is  0  or  any  positive  whole 
=  0,  this  expression  reduces  to  0 ;  if  n  is 
an  even  number,  say  2  m,  it  becomes  2  mir  +  0,  since  any  even 


GENERAL  VALUES  OF  ANTIFUNCTIONS  99 

power  of  —  1  is  +  1 ;  and  if  n  is  an  odd  number,  say  2to  +  1, 
the  expression  becomes  (2  m  +  1)  it  —  $,  since  any  odd  power 
of  -  1  is  -  1. 

By  giving  to  n  different  values,  0,  1,  2,  3,  •  •  • ,  we  can  find 
any  number  of  angles  having  the  same  sine  as  0. 

EXERCISES 

1.  Show  that  all  angles  having  a  given  cosine  are  included 
in  the  general  expression  2  nir  ±  6. 

2.  Show  that  all  angles  having  a  given  tangent  are  included 
in  the  general  expression  nir  -f-  0. 

3.  Name  five  angles  that  have  the  same  sine  as  a  given 
angle  0.  ^ 

4.  Name  six  values  of  cos-1-~-- 

m*     ,. .    .  .       „  ,  ,  sin  x       ,  tan  x 

71.    Limiting  Values  of  and  as  x  approaches  the 

x  x 

Value  Zero.  Let  A  OX  (Fig.  47)  be  any  positive  angle  less  than 
a  right  angle  and  let  x  be  the  number  of  radians  it  contains. 

To  prove  that  as  x  approaches  zero  as  a  limit, approaches 

1  as  its  limit.  With  0  as  a  center,  and 
with  any  convenient  radius,  describe  a 
circumference  cutting  OA  at  A  and  OX 
at  F.  Draw  A  C  perpendicular  to  OX  and 
produce  it  to  meet  the  circumference  at 
B.  Draw  the  tangents  AD,  BD.  Then, 
by  geometry, 

Z  BOX  =  Z  AOX,      AC  =  CB, 
AD  =  BD,       arc  AF  =  arc  BF. 

It  is  assumed  that  the  arc  AFB  is  less  than  AD  +  DB.     We 
have,  then, 

AF  +  FB  >  AC  +  CB,  but  <  AD  +  DB, 
or  2AF>2AC,  but  <2AD. 


100  PLANE    TRIGONOMETRY 

.\AF>  AC,     but  <AD. 

..  —  >—,    but<— • 

AF 
But,  by  formula  (41),  — —  equals  the  number  of  radians  in  the 

.       AF  ,       AC  , AD\ 

angle  AOX,  i.e.,  — —  =  a; ;   also,  — —  =  sin  x  ang.  -J—  fc=  tan  x. 
(J  A  (J  A  (J  A 

We  have,  therefore, 

x  >  sin  x,  but .  <  tan  x. 

Dividing  by  sin  x, 

x     ^i  v  4.  ^  tan  *  ^ 

-: >  1,        but  <  — : >  or 

sin  sc  sm  a;         cos  x 

But  the  greater  a  number  the  less  its  reciprocal;   .'. 

is  less  than  1,  but  greater  than  cos  x.     Now,  as  x  approaches 

,          ,-.-•-,          -,          sin  x 
the  limit  zero,  cos  x  approaches  the  limit  1,  and  as  

sin  X' 
always  lies  between  cos  x  and  1,  the  limit  of  is  also  1. 

tan  x      sin  x         1 
Again,  = X 


sm  x 


x  x         cos  x 

and    as  x  approaches   the   limit   zero,    and  both 

rr  x  cos  x 

approach  the  limit  1.      Therefore,  by  the  theory  of  limits, 
their  product,  X >  also  approaches  the  limit  1. 

r  X  COS  X 

*72.  Special  Solutions.  In  the  right  triangle  ACB  (Fig.  48) 
let  a  and  c  be  given.  If  they  are  nearly  equal,  A  is  nearly 
90°,  and  its  value  cannot  be  determined  with  the  usual  degree 
of  exactness  from  the  equation 

sm  A  =-• 
c 

For  example,  let  a  =  5235,  c  =  5237.1    By  the  usual  method, 
log  sin  A  =  log  a  —  log  c. 


SPECIAL   SOLUTIONS  101 


log  5235  =  13.71892  — 10 
log  5237.1  =    3.71909 
log  sin  A  =    9.99983  -  10 
And  on  referring  to  the  table,  we  find  it  impossible  to 
decide  whether  to  take  A  to  be  88°  21'  30",  88°  25'  10", 
or  some  intermediate  value.     In  such  cases,  the  follow- 
ing more  accurate  method  should  be  employed. 
First  find  £  A  by  formula  (27).     Thus,  we.  have 


/l  —  cos  A 

tan  \A—  \/t— 7 

2  *  1  +  cos  A 


or  tan  \  A 


=  V^t  ^3> 


Since  A  is  nearly  90°,  \  A  is  nearly  45°,  and  the  difficulty 
in  determining  A  is  obviated.     Before  applying  the  formula 
for  tan  \A,  however,  it  is  necessary  to  find  b.     This  is  done 
from  the  relation  b2  —  (c  +  a)  (c  —  a).     We  have 
c  +  a  =  10472.1. 
c-a  =  2.1. 
log  10472.1  =  4.02004 
log  2.1  =  0.32222 
2)4.34226 
log  b  =  2.17113 

b  =  148.3. 
c  -  b  =  5088.8. 
c  +  b  =  5385.4. 
log  5088.8  =    3.70661 
log  5385.4=    3.73122 

2)19.97539-20 
log  tan  \A  =    9.98770  -  10 
%A  =  44°  11' 19". 
A  m  88°  22'  38". 


102  PLANE   TRIGONOMETRY 

*  73.  The  same  difficulty  is  encountered  in  applying  the  law 
of  sines  in  Case  3  of  oblique  triangles,  when  the  unknown 
angle  opposite  one  of  the  given  sides  is  nearly  90°. 

For  example,  let  b  =  2,  a  =  1.06,  A  =  37°.     By  the  usual 

method, 

.     _      b  sin  A 


am  u  — 

a 

log  2=    0.30103 

log  sin  37°  =    9.72421  - 

-10 

10.02524  - 

-10 

log  1.06=    0.02531 

Fig.  49 


log  sin  B  =    9.99993-10 

And  we  cannot  decide  from  the  tables  whether  to  take  B  as 
88°  56'  10",  89°  0'  30",  or  some  inter- 
mediate value. 

To  obtain  the  value  of  B  more  accu- 
rately, we  may  proceed  as  follows : 
Draw  the  perpendicular  CD  =  p  (Fig. 
49),  and  let  DB  =  t.  We  first  deter- 
mine p  from  the  right  triangle  A  DC, 
then  t  from  the  relation  t2  =  a2  —  p2, 
and  then  apply  the  method  of  the  last  section  to  find  B  from 
the  right  triangle  CDB. 

Thus,  in  the  above  example,  we  have 

p  =  b  sin  A  =  2  sin  37°  =  1.05983. 
t2  =  (a  +  p)  (a-p)=  2.11983  x  0.00017. 
log  2.11983  =  10.32630  -  10 
log  0.00017  =    6.23045  -  10 
2  log  t  =  16.55675  -  20 
log  t  =  8.27837  -  10. 
t  =  0.01898. 


Then,  by  formula  (43),       tan  £  B  =  yj- =  \- 


04102 


+  t       M.07898 


SPECIAL   SOLUTIONS  108 

log  1.04102  =  20.01746  -  20 

log  1.07898  =    0.03301  ■ 

2)19.98445-20 
log  tan  \  B  =    9.99223-10 

%B  =  44°  29'  26". 
B  =  88°  58'  52". 

The  supplementary  value  91°  V  8"  must  also  be  accepted 
in  this  case  (see  Sec.  42),  giving  as  the  two  solutions  the 
triangles  ACBX,  ACB2.  Let  the  student  complete  the  solu- 
tions, finding  the  remaining  parts  for  each  triangle. 

Ans.  For  triangle  ACBU  Ct  =  54°    V    8",  cx  =  1.6187 ; 
for  triangle  ACB2,  C2  =  51°  58'  52",  c2  =  1.5758. 

EXERCISES 

1.  Given  a  =  999.7,  c  =  1000,  C  =  90°,  to  find  A. 

Ans.   A  =  $8°  35'  46". 

2.  Given  b  =  1,  c  =  26,  C  =  90°,  to  find  A. 

Ans.    A  =  87°  47'  46". 

3.  Given  a  =  148.228,  6  =  263,  ^4  =  34°  17',  to  find  B,  C,  c. 


*  CHAPTER    IX 


SURVEYING 


74.  The  fundamental  problem  in  land  surveying  is  the 
computation  of  the  area  of  a  tract  of  land  from  data  which 
determine  its  boundaries. 

Let  ABODE  (Fig.  50)  be  the  plot  of  a  farm.  The  bound- 
ing lines  AB,  BC,  etc.,  are  called 
courses.  The  usual  data  for  finding 
the  area  of  a  farm  are  the  lengths 
and  bearings  of  the  courses.  The 
lengths,  or  distances,  are  measured 
by  means  of  a  chain,  or  tape.  Gun- 
ter's  chain,  which  is  in  general  use, 
is  66  feet  long  and  is  divided  into 
100  links.  In  ordinary  surveying, 
bearings  are  measured  by  means  of 
a  compass,  an  instrument  by  which 
angles  can  be  read  to  quarter  degrees. 
For  more  accurate  work,  the  engineer's  transit,  reading  to 
minutes,  is  used.  The  field  notes  for  the  tract  plotted  in 
Fig.  50  are  as  follows : 


Fig.  50 


Cor  k  si: 

Beauinu 

Distance 

(chains) 

AB 
BC 
CD 
DE 
EA 

N.    40°  E. 
N.    60°  E. 
S.     20°  E. 
S.     60°  W. 
N.  38i°  W. 

7.18 

9.20 

15.22 

11.42 

12.72 

104 


SURVEYING  105 

75.  Units  of  Land  Measure.  The  units  of  land  measure 
commonly  used  are  as  follows : 

1  rod,  or  pole  =  16^  feet. 

1  chain  =  4  poles  =  66  feet. 
1  acre  =  10  square  chains. 

In  some  states  Spanish-American  units  are  in  use.  In  this 
system  the  unit  of  length  is  the  vara  [va'-ra].  In  Texas  a 
vara  is  taken  to  be  33£  inches ;  in  California,  33  inches.  The 
vara  used  in  Mexico  is  equivalent  to  32.9927  inches.  The  area 
of  a  square  whose  side  is  1000  varas  is  called  a  labor  [la-bor'] ; 
the  area  of  a  square  whose  side  is  5000  varas  is  called  a 
league. 

EXERCISES 

1.  How  many  square  poles  in  1  acre  ?  How  many  square 
feet  ?     How  many  square  yards  ? 

2.  How  many  acres  in  1  labor  ?    in  1  league  ? 

3.  How  many  varas  in  1  mile  ? 

76.  Areas  by  Coordinates.  In  Sec.  78  we  shall  show  how  to 
find  the  area  of  a  polygon  when  the  bearings  and  distances 
of  its  boundary  lines  are  given.  The  method  will  be  more 
readily  understood  after  the  solution  of  the  following. 

Problem.  To  find  the  area  S  of  a  polygon  in  terms  of  the 
coordinates  of  its  vertices. 

Let  the  coordinates  of  the  vertices  A,  B,  C,  D,  E  (Fig.  50) 
be  Xiyx,  x2y<i,  x3y3,  x±y±,  x5y5.  For  convenience  the  axes  are 
so  chosen  as  to  throw  the  polygon  entirely  in  the  first 
quadrant.  Drawing  the  perpendiculars  Aa  =  xlf  Bb  =  x2, 
etc.,  we  see  that  the  polygon  ABCDE  is  equivalent  to  the 
polygon  cCDEe  minus  the  sum  of  the  trapezoids  aABb,  bBCc, 
and  eEAa.  The  polygon  cCDEe  is  composed  of  the  trapezoids 
cCDd  and  dDEe. 

.'.  S  =  cCDd  +  dDEe  -  (aABb  +  bBcC  +  eEAa).         (a) 


106  PLANE    TRIGONOMETRY 

The  area  of  each  of  these  trapezoids  can  be  expressed  in 
terms  of  the  coordinates  of  its  vertices.  For  example,  con- 
sider the  trapezoid  cCDd.  Its  bases  cC  and  dD  are  equal 
respectively  to  x3  and  x4 ;  its  altitude  dc  is  equal  to  y3  —  y4. 
Its  area,  therefore,  is  £  (x3  -f  x4)  (y3  —  y4)  ;  so  for  the  other 
trapezoids. 

We  have,  therefore,  for  the  double  area  of  the  polygon, 
2  S  =  (x3  +  x4)  (y3  -  y4) 

+  (x4  +  xB)  (y4  -  y5)  —  [(«!  +  x2)  (y2  -  yx) 
+  (x2  +  xs)  (y3  -  y2)  +  (as,  +  xt)  (yx  -  y5)]. 
Performing   the    indicated    multiplications,    reducing    and 
rearranging,  we  have 

2S  =  y1(x2-  x5)  +  y2  (xs  -  xx) 

+  yz  {x4  -  x2)  +  y4  (x5  -  x3)  +  2/6  (xx  -  x4). 
The  proof  applies  to  a  polygon  of  any  number  of  sides ; 
hence,  to  find  the  double  area  of  a  polygon  when  the  coordi- 
nates of  its  vertices  are  given,  we  have  the  following 

Rule.  Multiply  the  ordinate  of  each  vertex  by  the  abscissa 
of  the  following  vertex  minus  the  abscissa  of  the  preceding 
vertex  and  add  the  products. 

EXERCISES 

1.  Find  the  area  of  a  polygon  having  its  vertices  at  the  fol- 
lowing points,  (12,  30),  (30,  48),  (60,  24),  (18,  6),  the  linear 
unit  being  one  chain.  Ans.  104.4  A. 

2.  Draw  the  triangle  which  has  its  vertices  at  the  points 
x\V\f  xiVi>  xsys,  and  find  its  area  by  the  method  of  this  section 
without  using  the  rule.     Find  it  also  by  the  rule. 

77.  Definitions.  If  the  y-axis  be  a  meridian  and  the  #-axis 
an  east-and-west  line,  the  ordinate  of  a  point  is  called  its 
latitude  and  the  abscissa  is  called  its  longitude.  In  Fig.  51 
let  BE  be  any  course,  and  iVS  the  meridian  through  its  initial 


SURVEYING 


107 


e 


Fig.  51 


point ;   through  its  terminal  point  E,  draw  EP  perpendicular 
to  NS.     Then  DP  is  the  difference  in  latitude  of  the  points  D 
and  E,  and  is  called  the  latitude-differ- 
ence of  the  course  DE.     PE  is  the  differ- 
ence in  longitude  of  the  points  D  and  E, 
and  is  called  the  longitude-difference  of 
the  course  DE.     Then,  from  the  figure, 
DP  =  DE  cos  NDE, 

and  PE  =  DE  sin  NDE. 

a 

That  is,  the  latitude-difference  of  a  course    q 
equals  the  length  of  the  course  multiplied 
by  the  cosine  of  the  bearing ;   the  longi- 
tude-difference equals  the  length  of  the  course  multiplied  by  the 
sine  of  the  bearing. 

When  the  latitude-difference  is  measured  northward  from 
the  initial  point  of  the  course,  it  is  called  a  northing  and  is 
taken  to  be  positive ;  when  it  is  measured  southward,  it  is  called 
a  southing  and  is  taken  to  be  negative.  Similarly,  longitude- 
differences  measured  eastward  are  called  eastings  and  are  taken 
to  be  positive,  and  those  measured  westward  are  called  west- 
ings and  are  taken  to  be  negative. 

The  double  longitude  of  a  course  is  the  sum  of  the  longitudes 
of  its  ends.  Thus,  in  Fig.  50,  the  double  longitude  of  AB  is 
Xi  -\-  x%. 

78.  Rule  for  finding  the  Area  of  a  Tract  when  the  Lengths  and 
Bearings  of  the  Courses  are  given.  In  the  expression  for  S  in 
equation  (a),  Sec.  76,  we  see  that  to  every  course  in  the  tract 
ABCDE  there  corresponds  a  trapezoid;  and  that  to  find  S  we 
add  the  trapezoids  cCDd  and  dDEe  corresponding  to  courses 
having  southerly  bearings,  and  from  their  sum  subtract  the 
sum  of  the  trapezoids  aABb,  bBCc,  and  eEAa  corresponding  to 
courses  having  northerly  bearings.  The  areas  of  the  first  set 
are  called  north  areas  ;   of  the  second  set,  south  areas.     Again, 


108 


PLANE   TRIGONOMETRY 


considering  the  expression  for  the  double  area  of  any  one  of 
these  trapezoids,  as  bBCc,  i.e.,  (x2  -f  xs)  (y3  —  y2),  we  see  that  it 
is  numerically  equal  to  the  product  of  the  double  longitude  of 
the  corresponding  course  by  its  latitude-difference.  To  find 
the  double  area  of  the  tract,  we  have,  therefore,  the  following 

Rule.  Multiply  the  double  longitude  of  each  course  by  its 
latitude-difference,  placing  the  product  in  a  column  of  north 
areas  when  the  bearing  is  northerly,  and  in  a  column  of  south 
areas  when  the  bearing  is  southerly.  Add  up  the  two  columns 
and  take  the  difference  of  the  results. 

79.  Rule  for  finding  the  Double  Longitudes.  Let  DE,  EF 
(Fig.  51)  be  any  two  successive  courses.  Then,  double  longi- 
tude of 

EF  =  eE  +fF  =  eE  +  dD  +  PE  +  TF. 

That  is,  the  double  longitude  of  any  course  equals  the  double 

longitude  of  the  preceding  course,  plus  the  longitude-difference 

of  that  course,  plus  the  longitude-difference  of  the  given  course. 

In  applying  this  rule,  remember  to  give  the  sign  —  to  westings. 

For  convenience  we  shall  take  the  reference  meridian  one 

chain  west  of  the  most  westerly  corner,  and 

suppose  the  survey  to  begin  at  that  corner, 

going  around  in  regular  order,  keeping  the 

tract  on  the  right. 

Then,  for  the  double  longitude  of  the  first 
course  AB  (Fig.  52),  we  have 

aA  +  bB  =  aA  +  bP  +  PB 
=  2  +  PB, 
since  a  A  —  bP  =  1. 

That  is,  the  double  longitude  of  the  first  equals 
2,  plus  its  longitude-difference. 
The  double  longitudes  of  the  remaining  courses  are  then 
found  by  the  rule  given  above. 


Fig.  52 


SURVEYING 


109 


80.  Computation  of  Latitude-Differences  and  Longitude-Differ- 
ences. Referring  to  the  field  notes  in  Sec.  74,  we  have,  for 
the  first  course  AB, 

lat.-diff.  =  7.18  cos  40°      =  7.18  x  0.7660  =  5.50. 
long.-diff.  =  7.18  x  sin  40°  =  7.18  x  0.6428  =  4.61. 

Making  similar  calculations  for  the  remaining  courses  and 
tabulating  the  results,  we  have 


Lat.- 

DlFF. 

Long 

-DlFF. 

N. 

S. 

E. 

w. 

AB 

N.  40°  E. 

7.18 

5.50 

4.(51 

BC 

N.  60°  E. 

9.20 

4.60 

7.97 

CD 

S.  20°  E. 

15.22 

14.30 

5.21 

BE 

S.  60°  W. 

11.42 

5.71 

9.89 

EA 

N.  38j°  W. 

12.72 

9.99 
20.09 

7.87 
17.76 

20.01 

17.79 

It  is  obvious  that  if  the  measurements  were  exact,  the  sum 
of  the  northings  would  equal  the  sum  of  the  southings,  and 
that  the  sum  of  the  eastings  would  equal  the  sum  of  the 
westings.  But  owing  to  the  impossibility  of  making  abso- 
lutely accurate  measurements,  this  will  not  usually  be  the 
case.     Thus,  in  the  above  example, 

sum  of  northings  =  20.09 

sum  of  southings  =  20.01 

error  =      .08 

sum  of  eastings  =  17.79 

sum  of  westings  =  17.76 

error  =      .03 

The  usual  practice  in  compass  surveys  is  to  balance  the 
work,  as  it  is  called,  by  distributing  the  errors  among  the 


110 


PLANE   TRIGONOMETRY 


several  courses  in  proportion  to  their  lengths.     Thus,  for  the 
case  in  hand,  the  perimeter  of  the  tract  is  55.74  chains. 


.*.  error  in  latitude  for  1  chain     = 


error  in  longitude  for  1  chain  = 


.08 
55.74 

.03 
55.74 


=  .0014. 


=  .0005. 


Multiplying  these  unit  errors  by  the  lengths  of  the  several 
courses,  we  have,  correct  to  two  places  of  decimals, 


Course 

Correction  in 

Correction  in 

Lat.-Diff. 

Long.-Diff. 

AB 

.01 

.00 

BC 

.01 

.00 

CD 

.02 

.01 

DE 

.02 

.01 

EA 

.02 
.08 

.01 
.08 

The  northings  are  too  great,  and  each  must  be  diminished 
by  the  corresponding  corrections ;  the  southings  are  too 
small,  and  each  must  be  increased ;  so  for  the  eastings  and 
westings. 

After  applying  these  corrections  the  work  stands  as  follows : 


Course 

Bearing 

DlST. 

Lat.-Diff. 

LONG.-DIFF. 

Balanced 

N. 

S. 

E. 

W. 

N. 

S. 

E. 

W. 

AB 
BC 
CD 
DE 
EA 

N.  40°  E. 
N.  60°  E. 
S.  20° E. 
S.  60°  W. 
N.38£°W. 

7.18 
9.20 
15.22 
11.42 
12.72 

5.50 
4.60 

9.99 
20.09 

14.30 
5.71 

4.61 
7.97 
5.21 

17.79 

9.89 

7.87 

5.49 
4.59 

9.97 
20.05 

14.32 
5.73 

4.61 
7.97 
5.20 

9.90 

7.88 

55.74 

20.01 

17.76 

20.05 

17.78 

17.78 

SURVEYING  111 

81.  Computation  of  Double  Longitudes.  Applying  the  method 
given  in  Sec.  79,  the  computation  of  the  double  longitudes  of 
the  several  courses  is  as  follows : 


long.-diff.  of  1st  course 

= 

4.61 

add 
double  longitude  of  1st  course 

__ 

2 

&6i 

long.-diff.  of  1st  course 

S3 

4.61 

long.-diff.  of  2d  course 

= 

7.97 

double  longitude  of  2d  course 

= 

19.19 

long.-diff.  of  2d  course 

== 

7.97 

long.-diff.  of  3d  course 

= 

5.20 

double  longitude  of  3d  course 

= 

32.36 

long.-diff.  of  3d  course 

= 

5.20 
37.56 

long.-diff.  of  4th  course 

=  - 

-    9.90 

double  longitude  of  Jfth  course 

= 

27.66 

long.-diff.  of  4th  course 

=  - 

-    9.90 

17.76 

long.-diff.  of  5th  course 

=   - 

-    7.88 

double  longitude  of  5th  course 

= 

088 

Check.  It  is  easy  to  show  from  a  figure  that  the  double 
longitude  of  the  last  course  is  numerically  equal  to  its  longi- 
tude-difference plus  2.  This  furnishes  a  check  on  the  accuracy 
of  the  work.  If,  as  is  frequently  done,  the  prime  meridian  is 
made  to  pass  through  the  most  westerly  corner,  double  longi- 
tude of  the  last  course  will  be  equal  numerically  to  its  longitude- 
difference. 

Inserting  the  above  values  for  the  double  longitudes  and 
computing  the  double  areas  by  the  rule  given  in  Sec.  78,  the 
work  takes  the  final  form  given  on  the  following  page. 


112 


PLANE    TRIGONOMETRY 


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SURVEYING 


113 


MISCELLANEOUS    EXERCISES 
1.  Find  the  area  of  the  tract  ABODE  from  the  following  field  notes 


Course 

Bearing 

Distance 
(poles) 

AB 
BC 
CD 
BE 
EA 

N.  41°  E. 
S.   70°  E. 
S.  29°  E. 

West 
N.  49}°  W. 

60 
40 

72 
75 
48 

Ans.  32.89-4. 


Course 

Bearing 

Distance 

(chains) 

AB 
BC 
CD 
DE 
EA 

N.  21°  E. 
N.  05°  E. 
S.  33°  E. 
S.  65}°  W. 
N.  48°  W. 

57.16 
7.68 
52.20 
50.96 
13.40 

Ans.   187.63  4. 


Course 

Bearing 

Distance 

(varas) 

AB 
BC 
CD 
BE 
EA 

N.  47°  E. 

East 

S.  32°  E. 

South 

N.  77°  30'  W. 

964 
822 
644 
523 
1913 

Ans.  235.79  acres. 


114 

4. 


PLANE    TRIGONOMETRY 


Course 

Bearing 

Distance 
(chains) 

AB 
BC 
CD 
DE 
EA 

S.  23°  E. 
S.  80°  E. 
N.  38°  E. 
N.  42°  W. 
S.  60|°W. 

48 
52 
46 
62 
64 

Ans.  499.82  acres. 


5.  Assuming  that  all  bearings  and  distances  were  accurately  measured, 
supply  the  bearing  and  distance  of  DE  in  the  following  : 


Course 

Bearing 

Distance 
(chains) 

AB 
BC 
CD 
DE 
EA 

South 
S.  88°  W. 
N.  42°  W. 

S.   18°  E. 

20 
18 
20 

18 

6.  Find  the  bearing  of  a  line  from  B  to  E  in  Exercise  1. 


SPHERICAL  TRIGONOMETRY 


CHAPTER   X 

82.  The  Spherical  Triangle.  The  portion  of  the  surface  of 
a  sphere  bounded  by  three  arcs  of  great  circles  is  called  a 
spherical  triangle.  The  bounding  arcs  are  called  the  sides  of 
the  triangle,  and  the  interior  spherical  angles  formed  by  the 
sides  are  the  angles  of  the  triangle.  Spherical  trigonometry 
is  concerned  with  the  relations  that  subsist  between  the  sides 
and  the  angles  of  a  spherical  triangle.  We  shall  consider 
only  triangles  in  which  each  side,  and  therefore  each  angle,  is 
less  than  180°. 

83.  Notation  and  Conditions.     Let  the  vertices  of  the  spher- 
ical triangle  ABC  (Fig.  53)  be  joined  to  the  center  of  the 
sphere,  0.     Denote  the  angles  of  b 
the  triangles  by  A,  B,  C  and  the                           ^^^^/X 
corresponding    sides    by    a,    b,    c.             ^^^^           J    \ 

Then  we  know,  by  geometry,  that    q<^__ / A~ 

the  angle  A  equals  the  plane  angle  **"*^k*^.      /^^ 

of  the  dihedral  angle  whose  edge  A 

is  OA  ;  so  for  B  and  C ;  and  that  FlG'53 

the  sides  a,  b,  c  measure  the  face  angles  BOC,  AOC,  BOA 

respectively. 

It  will  be  useful  to  recall  the  following  propositions,  which 
are  proved  in  geometry. 

In  any  spherical  triangle 

1.  The  greater  side  is  opposite  the  greater  angle,  and 
conversely. 

2.  The  sum  of  any  two  sides  is  greater  than  the  third  side. 

115 


116 


SPHERICAL    TRIGONOMETRY 


3.  The  sum  of  the  sides  is  less  than  360°. 

4.  The  sum  of  the  angles  is  greater  than  180°  and  less  than 
540°. 

Also  the  following : 

5.  If  A'B'C  be  the  polar  triangle  of  ABC,  ABC  is  the  polar 
triangle  of  A'B'C. 

6.  In  two  polar  triangles,  ABC  and  A'B'C', 

A  =  180°  -a',  B  =  180°  -V,   C  =  180°  -  c' ; 
A' =180° -a,    B'  =  180°-b,    C  =  180°  -  c . 

84.  Like  Parts  and  Unlike  Parts.  If  two  parts  of  a  spherical 
triangle  are  both  less  than  90°  or  both  greater  than  90°,  they 
are  said  to  be  like  parts ;  if  one  is  greater  than  90°  and  the 
other  less,  they  are  said  to  be  unlike. 

The  following  propositions  are  established  geometrically. 

1.  If  the  two  legs  of  a  right  spherical  triangle  are  like 
parts,  the  hypotenuse  is  less  than  90°  ;  if  the  two  legs  are 
tinlike  parts,  the  hypotenuse  is  greater  than  90°. 

2.  In  a  right  spherical  triangle  a  side  and  its  opposite  angle 
are  like  parts. 

Let  ACB  (Fig.  54)  be  a  right  spherical  triangle,  right  angled 

at  C,  having  the  legs  B  C  and 
AC  each  less  than  90°. 

Let  P  be  the  pole  of  the  arc 
A  C.  Then  CB  produced  passes 
through  P.  With  A  as  a  pole, 
and  a  quadrant  as  a  radius, 
describe  the  arc  PR,  cutting  A  C 
produced  in  R.  Then  PR  =  90° 
and  CP  =  90°.  Since  CB  is 
less  than  90°,  AB  produced  cuts 
Pit  in  some  point,  as  M,  between 
P   and   R.      Then   AM  =  90°. 


Fig.  54 


Hence,  the  hypotenuse  AB  is  less  than  90c 


LIKE    PARTS    AND    UNLIKE    PARTS 


117 


Again,  MR  is  the  measure  of  the  angle  A,  and  since  MR  is 
less  than  90°,  A  is  less  than  90°.  In  like  manner,  let  P'  be 
the  pole  of  the  arc  BC.  Then  CA  produced  passes  through  P'. 
With  5asa  pole,  and  a  quadrant  as  a  radius,  describe  the  arc 
P'R',  cutting  BC  produced  in  R'.  Then  P'R'  ==  90°.  Since  CA 
is  less  than  90°,  BA  produced  cuts  P'R'  at  some  point,  as  M', 
between  P'  and  R',  and  M'R',  the  measure  of  the  angle  B,  is 
less  than  90°.  Hence,  B  is  less  than  90°.  The  propositions 
are  true,  therefore,  for  the  case  considered. 

Next,  let  both  a  and  b  be  greater  than  90°,  as  in  Fig.  55. 
Complete  the lune CBC'A.     Then  AC  =  /-C,  and  in  the  right 


triangle  AC'B  each  leg  is  less  than  90°.  Hence,  the  hypote- 
nuse AB  is  less  than  90°,  and  the  angles  BAG',  ABC'  are  each 
less  than  90°;  therefore  their  supplements,  A  and  B,  are  each 
greater  than  90°. 

Next,  let  one  of  the  legs,  as  a  (Fig.  56),  be  greater  than  90°, 
and  the  other  leg,  b,  be  less  than  90°. 
Complete  the  lune  BAB'C.  Then  in  the 
right  triangle  ACB'  each  leg  is  less  than 
90° ;  therefore  the  hypotenuse  AB'  is  less 
than  90°,  and  its  supplement  AB,  the 
hypotenuse  of  ACB,  is  greater  than  90°. 
Also,  the  angles  B'  and  CAB'  are  each  less 
than  90°.  Hence,  B,  the  equal  of  B',  is 
less  than  90°,  and  A,  the  supplement  of  |l 
CAB',  is  greater  than  90°. 

Thus,  the  propositions  are  established 
for  all  cases. 


Fig.  56 


118  SPHERICAL   TRIGONOMETRY 

85.  The  Right  Spherical  Triangle.  Formulas.  Let  ACB 
(Fig.  57)  be  a  right  spherical  triangle,  C  being  the  right  angle, 
and  let  each  of  the  legs  a  and  b  be 
less  than  90°.  Join  each  of  the  ver- 
tices to  the  center  of  the  sphere,  O. 
From  any  point,  P,  in  OA  draw  PQ 
and  PR  perpendicular  to  OA,  in  the 
planes  OA B,  OAC  respectively.  Then 
QPR  is  the  plane  angle  of  the  dihe- 
dral angle  whose  edge  is  OA. 

.\Z  QPR  =  ZA. 

Join  Q  and  R.     Then,  by  geometry,*  QR  is  perpendicular 
to  the  plane  OA  C,  and  therefore  to  RP  and  OR. 
1.    We  have,  then, 


QR 
QR       OQ       sin  BOC 


sin  QPR  =  F 


.'.  sin  A  = 

Similarly,  sin  B  = 

*  The  proof  is  as  follows : 

1.  OA  is,  by  construction,  perpendicu 
dicular  to  the  plane  QRP. 

2.  .".  the  plane  OAC,  which  contains  OA,  is  perpendicular  to  the  plane  QRP. 

3.  Since  C  is  a  right  angle,  the  plane  OB C  is  also  perpendicular  to  the 
plane  OAC. 

4.  Since  the  intersecting  planes  OBC,  QRP  are  perpendicular  to  the  plane 
OAC,  their  intersection  QR  is  perpendicular  to  OAC. 

It  should  be  added  that  the  construction  of  the  right  triangle  QRP  is 
obviously  impossible  unless  the  dihedral  angles  whose  edges  are  OA  and  OB 
and  the  face  angle  QOP  are  each  less  than  90°.  But  by  the  last  article,  since 
a  and  b  are  each  less  than  90°,  A  and  B,  which  measure  these  dihedrals,  are 
each  less  than  90°,  and  the  hypotenuse  c,  which  measures  the  face  angle  QOP, 
is  also  less  than  90°. 


PQ       sin  BOA 
~OQ 

sin  a 
sin  c 

[i] 

sin  b 
sine 

[2] 

lar  to  PQ  and  PR. 

' .  OA  is  perpen- 

FORMULAS 

PR 

2. 

PR       OP 

cos  QPR  = = 5= 

PQ       PQ 

OP 

tan  b 

.'.  COS  ^4  = ;• 

tan  c 

tan  ^  OC 
tan  .4  OB 

Similarly, 

tan  a 

cos  jB  = 

tanc 

QR 

a 

„„„      QR       OR 

tfjn  OPT?  —           —  ...      — 

tan  £OC 

PR      PR 
OR 

tan  a 

.*.  tan  A  =  — ; — -- 
smo 

sin^OC 

Similarly, 

tan  5 

tan  B  = 

sin  a 

4. 

^„       OP       OP 

OR 

OQ 

- 

=  cos^40C  cos  BOC. 

.-.  cos  c  =  cos  a  cos  b. 

119 


5.  From  formula  [3],  we  have 


[3] 
M 


[5] 
[6] 


[7] 


sin  b      cos  c 
cos  ^4  = X  — 

cos  o      sin  c 

But  from  formula  [2], 

sin  b  =  sin  c  sin  B, 

and  from  formula  [7], 

cos  c  =  cos  «  cos  b. 

sine  sin  i?      cos  a  cos  ft 

.'.  cos  A  = ; —  X : > 

cos  b  sin  c 

or  cos  A  =  cos  a  sin  5.  [8] 

Similarly,      cos  B  =  cos  5  sin  v4.  [9] 


120  SPHERICAL   TRIGONOMETRY 


formulas  [8]  and 

m, 

cos  a 

cos  A 
sini? 

cos  b 

cos  B 

sin  A 

.*.  cos  a  cos  b 

cos  A 

=  — — ^  x 

cos 

B 

cot  A  cot  B. 
sin  B      sm  A 


.*.by  formula  [7],  cos  c  =  cot  A  cot  5.  [10] 

The  formulas  [1]  to  [10]  suffice,  as  will  be  seen  later,  for 

the  solution  of  right   spherical  triangles.      For   convenience 

they  are  here  collected. 


_.      A       sin  a 
sin  A  — 

[1] 

sm  b 
sin  if  =    . 

sine 

[2] 

sine 

tan  b 

cos  A  =  - 

tan  c 

[3] 

tan  a 

cos  B  = 

tan  c 

[4] 

tan  a 
tan  A  —  — — r- 
sm  b 

[5] 

tan  £ 

tan  2?  =  — 

sin  a 

[6] 

cos  c  =  cos  a  cos 

b. 

[7] 

cos  A  =  cos  a  sin  B. 

[8] 

cos  B  =  cos  b  sin 

A. 

[9] 

cos  e   =  cot  A  cot  jB. 

[10] 

86.  In  deriving  the  above  formulas  both  of  the  legs  a  and 
b  were  taken  to  be  less  than  90°.  The  proof  is  extended  so 
as  to  include  all  cases  as  follows : 

1.    Let  both  legs  be  greater  than  90°,  as  in  Tig.  55. 

Produce  the  arcs  CB,  CA  to  meet  in  C",  completing  the  lune 
CBC'A.  Then  C  =  C  =  90° ;  and  in  the  right  triangle  A  C'B, 
BC  =  180°  -  a,  AC'=>  180°  -  b.  Hence,  both  legs  are  less 
than  90°,  and  we  may  apply  to  this  triangle  the  formulas  of 
the  last  article. 

By  formula  [1],     sin  BA  C  =  S1.n  ^  > 
J  L   J'  sinJ3.1 


•    /iQAo       ,x      sin(180°-a) 
sin  (180   —  A  )  = *— ; L 


or 

sine 


FORMULAS  121 


t-     o        ^o                           .       sina 
.*.  by  Sec.  42,  sin  A  =  — > 

sm  c 

which  establishes  formula  [1]  for  this  case. 

Again,  by  formula  [3], 

n~„,      tan^C" 

cos£^4C"  = , 

tan  BA 

/iqao        ,s      tan  (180° -ft) 

or  cos  (180   —  A)  = ^ -- 

x  7  tan  c 

—  tan  ft 

.'.  —  cos  A  =  — j 

tanc 

tan  ft 
or  cos  A  = 

tan  c 

So  for  the  other  formulas. 

2.  Let  one  of  the  legs,  a,  be  greater  than  90°,  and  the  other, 
b,  be  less  than  90°,  as  in  Fig.  56. 

Produce  BC  and  BA  to  meet  at  B'.  Then,  in  the  right  tri- 
angle ACB',  B'  =  B,  CB'  =  180°  -  a,  AB'  =  180°  -  c. 

Here,  again,  both  legs  are  less  than  90°,  and  we  have,  by 

formula  [1],  . 

•  sm  B'C 

smB'AC  ==-. — — ,i 
sm  AB' 

■     /iQAo       A^      sin  (180° -a) 

or  sin  (180   —  A)=    .    v/iQAO f- 

v  y      sm  (180   —  c) 

sin  a 

.'.  sm  yl  as  — 

sin  c 

Again,  by  formula  [3], 

tan  A  C 
cos  B'A  C  = 


or  cos  (180°  -  A)  = 

. ' .  cos  A  = 


tan  AB' 

tan  ft 
tan  (180°  —  c) 
tan  ft 


tanc 
So  for  the  other  formulas. 


122  SPHERICAL   TRIGONOMETRY 

87.  Napier's  Rules.     It  is  difficult  to  retain  the  formulas  of 
the  last  article  in  the  memory,  and  the  necessity  for  doing  so 
is  obviated  by  the  following  device, 
|eo"       known  as  Napier'' s  Rules. 

Corresponding  to  the  right  spherical 
triangle  A  CB  construct  a  diagram  let- 
tered as  in  Fig.  58,  where  a  and  b  have 
the  same  values  as  in  the  given  triangle 
ACB,  and  where  co-^1,  co-B,  co-c  mean 
complement  of  A,  complement  of B,  complement  ofc  respectively. 
The  five  parts,  co-^4,  b,  a,  co-B,  co-c,  occur  in  the  figure  in  the 
same  order  in  which  the  parts  A,b,  a,  B,c  occur  in  the  triangle 
A  CB,  and  are  called  the  circular  parts  of  the  triangle  A  CB. 
Note  that  the  right  angle  C  is  not  included  among  the  circular  parts. 

If  any  one  of  these  five  circular  parts  be  considered  as  a 
middle  part,  the  two  parts  lying  next  to  it  are  called  adjacent 
parts,  and  the  remaining  two  parts  are  called  opposite  parts. 

Thus,  if  co-^4  be  taken  as  a  middle  part,  the  adjacent  parts 
are  b  and  co-c ;  and  the  opposite  parts  are  a  and  co-B. 

Napier's  Rules  are  as  follows : 

I.  The  sine  of  a  middle  part  equals  the  product  of  the  tan- 
gents of  the  adjacent  parts. 

II.  The  sine  of  the  middle  part  equals  the  product  of  the 
cosines  of  the  opposite  parts. 

These  rules  may  be  easily  remembered  by  noting  the  occur- 
rence of  the  letter  a  in  tangents  and  adjacent,  and  of  the 
letter  o  in  cosines  and  opposite. 

The  rules  are  verified  by  considering  each  of  the  five  circu- 
lar parts  in  turn  as  below : 

1.  If  we  consider  the  circular  part  a,  the  adjacent  parts  are 
b  and  co-B. 

According  to  Rule  I, 

sin  a  =  tan  b  tan  (co-B), 


NAPIER'S   RULES  123 

or,  since  the  tangent  of  the  complement  of  B  equals  cot  B, 

sin  a  rs  tan  b  cot  B ; 
and  this  equation  is  true  by  formula  [6]. 
The  opposite  parts  are  co-^4,  co-c. 
According  to  Rule  II, 

sin  a  =  cos  (co-A)  cos  (co-c), 
or  sin  a  =  sin  A  sin  c, 

which  is  true  by  formula  [1]. 

2.  Consider  next  the  circular  part  b.     The  adjacent  parts 
are  a  and  co-^4. 

According  to  Rule  I, 

sin  b  =  tan  a  tan  (co-A), 
or  sin  b  =  tan  a  cot  A, 

which  is  true  by  formula  [5]. 
The  opposite  parts  are  co-c,  co-U. 
According  to  Rule  II, 

sin  b  =  cos  (co-c)  cos  (co-5), 
or  sin  b  =  sin  c  sin  B, 

which  is  true  by  formula  [2]. 

3.  Consider  next  the  circular  part  co-A     The  adjacent  parts 
are  b  and  co-c. 

According  to  Rule  I, 

sin  (co-^4)  =  tan  b  tan  (co-c), 
or  cos  A  =  tan  b  cot  c, 

which  is  true  by  formula  [3]. 

The  opposite  parts  are  a  and  co-B. 
According  to  Rule  II, 

sin  (co-^4)  =  cos  a  cos  (co-£), 
or  cos  A  =  cos  a  sin  B, 

which  is  true  by  formula  [8]. 

Let  the  student  complete  the  verification  by  considering  the  circular  parts 
co-c  and  co-jB  in  turn  as  middle  parts. 


124  SPHERICAL    TRIGONOMETRY 

88.  Solution  of  Right  Spherical  Triangles.  If,  besides  the 
right  angle,  any  two  parts  of  a  right  spherical  triangle  be 
given,  the  remaining  parts  can  be  found  by  applying  Napier's 
Rules.  The  Rules  should  be  so  applied  that  each  equation 
formed  shall  contain  only  one  unknown  quantity.  In  any 
case  the  solution  may  be  effected  by  the  following  steps  : 

1.  Take  one  of  the  given  parts  as  a  middle  part  and  apply 
to  it  that  one  of  Napier's  Rules  which  involves  the  other 
given  part. 

2.  Take  the  other  given  part  as  a  middle  part  and  apply 
the  same  rule. 

The  equations  thus  formed  determine  two  of  the  required 
parts.     Then, 

3.  Take  the  third  required  part  as  a  middle  part  and  again 
apply  the  same  rule. 

4.  As  a  check  take  the  part  last  found  as  a  middle  part  and 
apply  the  other  rule. 

The  manner  of  applying  these  rules  will  be  illustrated  by 
examples. 

Before  beginning  a  solution  the  student  should  draw  a  figure  showing  the 
five  circular  parts,  as  in  Sec.  87. 

EXAMPLES 

1.    Given  A  =  98°  16',  b  =  57°  14',  to  find  a,  c,  B. 

Taking  the  circular  part  co-A  as  a  middle  part  and  applying  Rule  I, 

we  have 

sin  {go- A)  =  tan  o  tan  (co-c), 
—  +        - 

or  cos  A  =  tan  b  cot  c. 

Since  A  >  90°,  cos  A  is  negative  ;  since  6  <  90°,  tan  6  is  positive. 
..-.  cot  c  is  negative.  Hence,  c  is  greater  than  90°,  and  we  must  take  as  the 
value  of  c  the  supplement  of  the  angle  found  in  the  table.  It  will  be 
found  convenient  to  write  the  algebraic  signs  over  the  functions,  as  above. 
Applying  logarithms  to  the  above  equation,  we  have 
log  cot  c  =  log  cos  A  —  log  tan  b. 


RIGHT    SPHERICAL   TRIANGLES  125 

log  cos  98°  16'  =  19.15770  -  20 
log  tan  57°  14'  =  10.19136  -  10 

log  cot  c=    8.96634-10 

180°  -  c  =  84°  42'  46//. 
c  =  95°  17'  14". 

Next,  apply  the  same  rule  to  the  part  6,  and  we  have 

+  —        — 

sin  b  =  cot  A  tan  a. 

log  tan  a  =  log  sin  b  —  log  cot  A. 
log  sin  57°  14'  =  19.92473  -  20 
log  cot  98°  16'  =    9.16224-  10 

log  tan  a  =  10.76249-  10 

180°  -a  =  80°  11'  49". 
a  =  99°  48'  11". 

Next,  apply  the  same  rule  to  the  remaining  part,  co-2?,  and  we  have 

+  —       — 

cos  B  =  tan  a  cot  c. 

log  cos  B  =  log  tan  a  +  log  cot  c. 
log  tan  a  =  10.76249-  10 
log  cote   =    8.96634  -  10 
log  cos  B=    9.72883-10 
B  =  57°  37/. 
Check.     Apply  the  other  rule  to  the  part  last  found,  giving 
cos  B  =  cos  b  sin  A. 
log  cos  B  —  log  cos  b  +  log  sin  A. 
log  cos  57°  14'  =  9.73337  -  10 
log  sin  98°  16'  =  9.99546  -  10 
log  cos  B  =  9.72883-  10 
which  agrees  with  the  value  found  above  for  log  cos  B. 

2.    Given  a  =  97°  22',  b  =  46°  18',  to  find  A,  B,  c. 

Applying  Rule  I  to  the  part  a, 

+  +        + 

sin  a  =  tan  b  cot  B. 

log  cot  B  =  log  sin  a  —  log  tan  6. 
log  sin  97°  22'  =  19.99640  -  20 
log  tan  46°  18'  =  10.01971  -  10 

log  cot  B=    9.97669  -  10 
B  =  46°  32'  12". 


126  SPHERICAL   TRIGONOMETRY 

Applying  the  same  rule  to  the  part  o, 

4-  —  — 

sin  b  =  tan  a  cot  A. 
log  cot  A  =  log  sin  b  —  log  tan  a. 
log  sin  46°  18'  =  19.85912  -  20 
log  tan  97°  22'  =  10.88845  -  10 
log  cot  A  =    8.97067-10 
180°  -  A  =  84°  39'  36". 
A  =  95°  20'  24". 

Applying  the  same  rule  to  the  remaining  circular  part,  co-c, 

—  —         + 

cos  c  =  cot  A  cot  B. 

log  cos  c  =  log  cot  A  +  log  cot  B. 

log  cot  A  =  8.97067  -  10 

log  cot  B  =  9.97669  -  10 

log  cose   =8.94736-  10 

180°  -  c  =  84°  55'  4". 

c  =  95°  4'  56'. 

As  a  check,  apply  Rule  II  to  the  part  just  found  : 

cos  c  =  cos  a  cos  b. 

log  cos  c  =  log  cos  a  +  log  cos  b. 

log  cos  a  =  9.10795  -  10 

log  cos  b  =  9.83940  -  10 

log  cose  =8.94735-  10 

agreeing  with  the  ahove  value  of  log  cos  c  within  one  point. 

3.   The  ambiguous  case.      Given  a  =  64°  29',  A  =  78°  10', 

to  find  b,  c,  B. 

+  +       + 

By  Rule  II,  sin  a  =  sin  A  sin  c, 

sin  a 
or  sin  c  = 


sin  A 

As  c  is  to  be  found  from  its  sine,  there  are,  by  Sec.  42,  two  solutions  ; 

namely,  the  angle  found  in  the  table,  and  its  supplement. 

Again,  applying  the  same  rule  to  co-^4, 

cos  A  =  cos  a  sin  B, 

.     _      cos  A 

or  sin  B  = —  • 

cos  a 

And  to  find  6,  we  have,  by  the  same  rule, 

sin  b  =  sin  c  sin  B  ; 


RIGHT   SPHERICAL   TRIANGLES  127 

so  that  B  and  b  also  are  to   be  found  from  their  sines,  and  the  same 
ambiguity  occurs  in  regard  to  them. 

It  is  easily  seen  geometrically,  also,  that  there  are  two  solutions. 
For,  let  A  CB  (Fig.  59)  be  a  right  spherical  triangle  having  the  given  parts 
A  and  a.  Then,  on  completing  the 
lune  ABA'C,  we  form  a  second  tri- 
angle, A'CB,  which  also  fulfills  the 
given  conditions ;  for  A'  —  A,  BC  =  a, 
and  Z  BCA'  is  a  right  angle.  It  is 
evident  also  from  the  figure  that  the  Fig.  59 

required  parts  BA',  CA',  A  CBA'  are 
the  supplements  of  the  corresponding  required  parts  in  the  triangle  A  CB. 

The  case  in  which  a  side  and  the  opposite  angle  are  the  given  parts  is 
accordingly  called  the  ambiguous  case. 

To  complete  the  solution,  apply  logarithms  to  the  equations  stated 
above.     Thus, 

log  sin  c  =  log  sin  a  —  log  sin  A. 

log  sin  64°  29'  =  19.95543  -  20 
log  sin  78°  10'  =    9.99067  -  10 
log  sin  c=    9.96476-10 
cx  =  67°  13'  48". 
c2  =  112°  46'  12". 
Also,  log  sin  B  =  log  cos  A  —  log  cos  a.„ 

log  cos  78°  10'  =  19.31189  -  20 
log  cos  64°  29'  =    9.63425  -  10 
log  sin  B=    9.67764-10 
Bt  =  28°  25'  37". 
B2  =  151°  34'  23". 
Also,  log  sin  b  =  log  sin  B  +  log  sin  c. 

logsinjB  =  9.67764-  10 
log  sin  c  =  9.96476  -  10 
log  sin  b  =  9.64240  -  10 
6i  =  26°  2'  9". 
62  =  153°  57'  51". 

The  two  solutions,  therefore,  are 

1.  &i  =  26°  2'  9",         Bx  =  28°  25'  37",     cx  =  67°  13'  48". 

2.  b.2  =  153°  57r  51",  B2  =  151°  34'  23",  c2  =  112°  46'  12". 
Check,     sin  6  =  tan  a  cot  A. 


128  SPHERICAL   TRIGONOMETRY 

4.    Given  a  =  105°  18',  c  =  98°  9',  to  find  b,  A,  B. 

By  Rule  II,  sin  a  =  cos  (co-c)  •  cos  (co-yl), 

+  +       + 

or  sin  a  =  sin  c  sin  A. 

log  sin  A  =  log  sin  a  —  log  sin  c. 
log  sin  105°  18'  =  19.98433  -  20 
log  sin  98°  9'  =    9.99559  -  10 
log  sin  A=    9.98874-10 
Since  a  >  90°,  A  >  90°,  and  we  must  use  as  the  value  of  A  the  supple- 
ment of  the  angle  found  in  the  table. 

.-.  180°  -  A  =  77°  0'  40,/. 
A  =  102°  59'  20". 
Applying  the  same  rule  to  the  circular  part  co-c, 

sin  (co-c)  =  cos  a  cos  6, 
—  —        + 

or  cos  c  =  cos  a  cos  b. 

log  cos  b  =  log  cos  c  —  log  cos  a. 
log  cos  98°  9'  =  19.15157  -  20 
log  cos  105°  18'  =    9.42140  -  10 
log  cos  b=    9.73017-10 
b  =  57°  30'  15". 
Applying  the  same  rule  to  the  remaining  part, 

sin  (co- J?)  =  cos  b  cos  (co-,4), 
+  +        + 

or  cos  B  —  cos  6  sin  .4. 

log  cos  B  =  log  cos  b  +  log  sin  A. 
log  cos  b  =  9.73017  -  10 
log  sin  A  =  9.98874  -  10 
log  cos  5=9.71891  -  10 
B  =  58°  21'. 
Write  down  a  check  formula  and  apply  it. 

Instead  of  the  use  of  algebraic  signs  to  determine  whether 
a  part  of  a  right  spherical  triangle  is  greater  or  less  than  90°, 
as  in  this  section,  we  may  apply  the  principles  of  like  parts 
and  unlike  parts  (Sec.  84).  Thus  in  Example  1,  since  A  >  90° 
we  have  a  >  90 ;  also,  a  and  b  being  unlike  parts,  c  must  be 
greater  than  90°;  again,  since  b  <  90°  we  must  have  also 
B  <  90°. 


RIGHT   SPHERICAL   TRIANGLES  129 

EXERCISES 

Solve  the  following  right  spherical  triangles,  checking  the 
work  where  the  answers  are  not  given  : 

1.  Given  a  =  48°  15',  c  =  74°  12',  to  find  A,  b,  B. 

Ans.  A  =  50°  50'  12",  b  =  65°  51'  51",  B  =  71°  30'  57". 

2.  Given  a  =  70°  28',  c  =  98°  18',  to  find  A,  b,  B. 

Ans.  A  =  72°  15'  15",  b  =  115°  34'  42",  B  =  114°  16'  50". 

3.  Given  A  =  36°  20',  c  =  81°  24',  to  find  b,  B,  a. 
Ans.  b  =  79°  22'  3",  B  =  42°  16'  48",  a  =  35°  51'  39". 

4.  Given  A  =  53°  23',  c  =  108°,  to  find  b,  B,  a. 

Ans.  b  =  118°  34'  46",  B  =  112°  34'  45",  a  =  49°  45'  36". 

5.  Given  a  =  32°  20'  40",  B  =  56°  20'  20",  to  find  b,  c,  A. 
Ans.  b  =  38°  46'  42",  c  =  48°  48'  19",  A  =  45°  19'. 

6.  Given  a  =  122°  15',  B  =  14°  20',  to  find  b,  c,  A. 
Ans.  b  =  12°  11'  38",  e  =  121°  26'  19",  A  =  97°  35'  28". 

7.  Given  a  =  56°  13',  b  =  34°  22',  to  find  B,  A,  c. 

Ans.  B  =  39°  26'  51",  A  =  69°  18'  39",  c  =  62°  40'  43". 

8.  Given  a  =  68°  27',  b  =  108°  40',  to  find  B,  A,  c. 
Ans.  B  =  107°  26'  37",  A  =  69°  29'  13",  c  =  96°  45'  5". 

9.  Given  A  =  76°  15',  B  =  49°  3',  to  find  a,  b,  c. 

Ans.  a  =  71°  39'  27",  b  =  47°  33'  56",  c  =  77°  44'  26". 

10.  Given  A  =  106°,  B  =  64°,  to  find  a,  b,  c. 

Ans.  a  =  107°  51'  32",  b  =  62°  52'  5",  e  =  98°  2'  22". 

11.  Given  A  =  32°  10',  a  =  24°  8',  to  find  B,  c,  b. 

Ans.  1.  5!  =  68° 3' 36",      cx  =  50°  10' 27",    b1  =45°25'45". 
2.  B2  =  111°  56'  24",  c2  =  129°  49'  33",  b2  =  134°  34'  15", 

12.  Given  A  =  105°,  a  =  134°,  to  find  B,  c,  b. 

Ans.  1.  -Bx  =  21°  52'  31",  cx  =  48°  8'  5",        h  =  16°  6' 34". 
2.  B2  =  158°  7'  29",  c2  =  131°  51'  55",  52  =  163°  53'  26". 


130 


SPHERICAL    TRIGONOMETRY 


13.  Given  B  =  60°,  b  =  50°,  to  find  A,  c,  a. 

Ans.  1.  ^  =  51°  3' 54",    Cl  =  62°  11' 43",    ax  =  43°  28'  32". 
2.  A%  =  128°  56'  6",  c2  =  117°  48'  17",  a2  =  136°  31'  28". 

14.  Given  B  =  100°,  b  =  106°,  to  find  A,  c,  a. 

Ans.  1.  Ax  =  39°  2'  56",    e,  =  102°  33'  20",  ax  =  37°  56'  42". 
2.  A2  =  140°  57'  4",  c2  =  77°  26'  40",    a2  =  142°  3'  18". 

15.  Given  A  =  110°  26',  B  =  34°  19',  to  find  a,  b,  c. 

16.  Given  A  =  68°  12',  c  =  73°  31',  to  find  b,  B,  a. 

17.  Given  B  =  106°,  6  =  112°  12',  to  find  A,  a,  c. 
18/  Given  a  =  72°  18',  b  =  103°,  to  find  B,  A,  c. 

19.  Given  a  =  53°  24',  c  =  76°  12',  to  find  A,  b,  B. 

20.  Given  a  =  38°  16',  B  =  69°  12',  to  find  b,  c,  A. 

89.  Quadrantal  Triangles.  A  quadrantal  triangle  is  one  in 
which  one  side  is  a  quadrant.  The  polar  triangle  of  such  a 
triangle  is  a  right  triangle.  Why?  After  solving  the  polar 
triangle,  the  parts  of  the  given  triangle  are  readily  found  by 
Sec.  83,  proposition  6. 


EXERCISES 

1.  Given  a  =  66°  32',  b  =  59°  43',  c  =  90°, 

to  find  C  =  104°  41'  7",  A  =  62°  32'  20",  B  =  ? 

2.  Given  c  =  90°,  B  =  74°  45',  a  =  18°  12', 

to  find  C  =  104°  31'  13",  &  =  83°  17'  15",  A  =  17°  35'  57". 

90.  Isosceles  Spherical  Triangles.  If,  in 
a  spherical  triangle  ABC  (Fig.  60),  we 
have  AB  =  A  C,  then,  by  geometry, 

ZB  =  ZC, 
and  AD,  the  arc  of  a  great  circle  drawn 
from  A  to  the  mid-point  of  BG,  divides 
the  triangle  ABC  into  two  equal  right 
triangles,  each  of  which  can  be  solved  by 
Kapier's  Rules. 


OBLIQUE   SPHERICAL   TRIANGLES 


131 


EXERCISES 

1.  Given  b  =  c  =  81°  24',  ^L  =  72°  40', 

to  find  a  =  71°  43'  18",  B  =  C  =  42°  16'  48". 

2.  Given  &  =  c  =  74°  13',  ^4  =  98°,  find  the  remaining  parts 
and  check  the  results. 

PROPERTIES   OF   SPHERICAL   TRIANGLES 

Oblique  spherical  triangles  are  solved  by  means  of  certain 
relations  which  subsist  between  the  sides  and  the  angles. 
These  will  now  be  established. 

91.  The  Law  of  Sines.  In  any  spherical  triangle  the  sines  of 
the  sides  are  to  each  other  as  the  sines  of  the  opposite  angles. 

In  the  oblique  spherical  triangle  ABC  draw  the  arc  BD 
perpendicular  to  A  C.     In  Fig.  61,  D  falls  between  A  and  C ; 


in  Fig.  62,  D  falls  on  CA  produced.     By  Napier's  Eule  II,  we 
have,  from  the  right  spherical  triangle  ABB  in  Fig.  61, 

sin  BD  =  sin  c  sin  A  ; 
in  Fig.  62,  sin  BD  =  sin  c  sin  (180°  —  A)  =  sin  c  sin  A. 

Similarly,  from  the  right  triangle  BDC  in  either  figure, 

sin  BD  =  sin  C  sin  a. 


lo2  spiii:::;cal  t::igoxometry 


• 

".  sin  C  sin  a  =  sin  c  sin  A 

sin  a       sin  A 

sin  c       sin  C 

Similarly, 

sin  a       sin  .4 
sin  6      sini? 

and 

sin  b       sin  2? 

sm  c      sm  C 
From  these  equations,  we  have 

sin  a       sin  b        sin  0 


[ii] 


sin  A       sin  i?      sin  C 

92.  Law  of  Cosines.  In  a  spherical  triangle  the  cosine 
of  any  side  equals  the  product  of  the  cosines  of  the  other 
two  sides  plus  the  product  of  their  sines  and  the  cosine  of 
their  included  angle,  or,  in  symbols,  taking  the  side  a  of  the 
triangle  ABC, 

cos  a  =  cos  b  cos  c  -f  sin  b  sin  c  cos  A.        [12] 

Draw  the  arc  BD  perpendicular  to  A  C. 
1.    Let  D  fall  between  A  and  C,  as  in  Fig.  61.     From  the 
right  triangle  BDC,  we  have,  by  Eule  II, 

cos  BC  =  cos  BD  cos  DC, 
or  cos  a  =  cos  BD  cos  (b  —  AD) 

=  cos  BD  (cos  b  cos  AD  +  sin  &  sin  yl  D) 
=  cos  BD  cos  /1Z>  (cos  &  +  sin  b  tan  ^4D). 
Now,  from  the  right  triangle  ADB,  by  Rule  II, 
cos  BD  cos  ^4  Z)  =  cos  c ; 
and,  by  Rule  I,  cos  A  =  cot  c  tan  yl  D, 
from  which       tan  AD  =  tan  c  cos  .1. 

Substituting  these  values  of  cos  BD  cos  AD  and  tan  vlZ)  in 
the  expression  for  cos  a,  we  have 

cos  a  =  cos  c  (cos  &  +  sin  b  tan  c  cos  A). 
.'.  cos  a  =  cos  b  cos  c  -f-  sin  b  sin  c  cos  A. 


OBLIQUE    SPHERICAL    TRIANGLES  133 

2.    Let  D  fall  on  CA  produced,  as  in  Fig.  62.     Then,  pro- 
ceeding as  before,  we  have,  from  the  right  triangle  BDC, 
cos  a  =  cos  BD  cos  CD 

=  cos  BD  cos  (b  -f  AD) 
=  cos  BD  (cos  b  cos  AD  —  sin  b  sin  ^4D) 
=  cos  BD  cos  y4Z>(cos  &  —  sin  b  tan  ^4Z>). 
But    cos  BD  cos  ^4D  =  cos  c, 
and  cos  5.4 D  =  cot  c  tan  AD. 

But  cos  5.4/;  =  cos  (180°  -  A)  =  -  cos  A 

.".  —  cos  A  =  cot  c  tan  ^4D, 
from  which  tan  AD  =  —  tan  c  cos  ^4. 
Substituting  as  before, 

cos  a  =  cos  c  (cos  6  -+-  sin  b  tan  c  cos  A), 
or  cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A. 

Similarly,      cos  b  =  cos  a  cos  c  +  sin  a  sin  c  cos  B. 
cos  c  =  cos  a  cos  &  +  sin  a  sin  &  cos  C. 

93.    To  express  sin  ^  A  in  terms  of  functions  of  a,  b,  c.     By 

formula  (25),  we  have 

2  sin2i  A  =  1  —  cos  A. 

From  formula  [12], 

cos  a  —  cos  b  cos  c 

COS  vl  = : — : — : 

sin  b  sin  g 

Substituting  this  value, 

rt    .  - ,  _       cos  a  —  cos  b  cos  c 

2  sm2  i  ^4  =  1 : — —. 

sin  b  sm  c 

_  sin  b  sin  c  +  cos  6  cos  c  —  cos  a- 

sin  b  sin  c 

_  cos  (b  —  c)  —  cos  a 

sin  6  sin  c 

_  2  sin  ^  (a  +  b  —  c)  sin  \(a  —  b  +  c) 

sin  6  sin  c 

by  formula  (18). 


134  SPHERICAL   TRIGONOMETRY 

Let         a-\-b  +  c  —  2  s. 
Then,     a  +  b-c  =  2s-2c  =  2(s-c). 
a-b  +  c  =  2s-2b  =  2(s-b). 

Substituting,  we  have 

2  sin  (s  -  c)  sin  (s  —  b) 

2  sin2  \  A  = *-^ — -r—. — » - 

z  sin  b  sm  c 


/sin(s  —  b)  sin  (s  —  c) 

.:&miA=\ K     ■    !    .     -;'  [13] 

■  *  sm  b  sm  c  L     J 

-....,        .                    /sin  (s  —  a)  sin  (s  —  c) 
Similarly,  sm  £  B  =  -^ 


sin  a  sm  c 


_         /sin  (s  —  a)  sin  (s  —  b) 

sin  ±  C  =  -V * — : — fr- 7 * 

'  *  sm  a  sm  t> 


94.    To  express  cos  \  A  in  terms  of  functions  of  a,  b,  c.     By 

formula  (26), 

2  cos2  £  4  =  1  +  cos  A 

cos  a  —  cos  J  cos  c 
sin  6  sin  c 
cos  a  —  cos  b  cos  c  -f-  sin  b  sin  c 


sin  6  sin  c 
_  cos  a  —  cos  (b  +  c) 

sin  b  sin  c 
_  2  sin  ±(a  +  6  +  c)sin%  (b-\-c— a) 
sin  6  sin  c 

by  formula  (18). 

/sin  s  sin  (5  —  a)  _.._. 

'.cos^^=\ ^T^ L'  C14] 

2  '       sin  i  sm  c  u 


„.     .,     ,                          /sin  5  sin  (s  —  6) 
Similarly,  cos  15  =  V : • L ' 

**  *  *         ein  n  sin  n. 


sm  a  sm  c 


sinssins-c) 

cos  1  C  =  \ : *! — r-1  - 

J  *      sin  a  sm  b 


OBLIQUE    SPHERICAL    TRIANGLES  135 

95.    To  express  tan  £  A  in  terms  of  functions  of  a,  b,  c.     From 
formulas  [13]  and  [14],  by  division  and  reduction, 


i  A     ^  /sm  (s  —  &) sm  0  — c)  rlK1 

tan  £  A  =  \ \ /—— * — -^  •  [15] 

1      sm  s  sm  (s  —  a)  L     J 

n-     •■!     i  i  ^         /sm  (5  —  a)  sin  (s  —  c) 

Bumkrly, tan ^  =  V     s\n s sin (g  1 ,)    *' 

■  *       sm  s  sm  (s  —  c) 

As  in  the  corresponding  case  in  plane  trigonometry,  these 
formulas  may  be  modified  as  follows  : 

In  the  formula  for  tan  £  A,  multiply  numerator  and  denomi- 
nator of  the  fraction  under  the  radical  by  sin  (s  —  a),  and  we 

have  __^ 

tan         _^  Jsin  (s  -  a)  sin  (s  -  b)  sin  Q  -  e) 
*  sin  s  sin2  (s  —  a) 


1  /sin  (s— a)  sin  (s—b)  sin  (s— c) 

sin (s— a)  * 


(5— a)  *  sin  5 

Now,  let  r  =  ■>*(»  -  a) am («  -  S) sin («  -  «)  _ 

'  sm  s 

Then,         tan  *  4  =  -— r  •  [16] 

Similarly,  tan  ^B  — 
tan  %  C  = 


sin  (5  —  b) 

r 
sin  (s  —  c) 


96.    To  express  the  sines,  cosines,  and  tangents  of  the  half  sides 
in  terms  of  functions  of  the  angles. 

1.   To  express  sin  \  a  in  terms  of  functions  of  A,  B,  C. 
We  have  a  =  180°  -A'. 

.-.^a  =  90°-^A'. 
.'.  sin  \a  =  cos  \ A'. 


136  SPHERICAL   TRIGONOMETRY 

But  in  the  polar  triangle  A'B'C,  we  have,  by  formula  [14], 


/sin  s'  sin  (V  —  a') 

gos^A'  =\ : — 77^ — L> 

2  ^       sin  0'  sm  c' 

where      a'  +  5'  +  c'  =  2  5'. 

We  have  now  to  express  the   quantity  under  the  radical 
sign  in  terms  of  functions  of  A,  B,  C. 
We  have  a'  =  180°  -  A 

b'  =  180°  -  B 

c'  =  180°  -  C 

.-.  a'  +  b'  +  6''  =  540°  -  (,-1  +  £  +  C). 

Let     4  +  ^  +  C  =  2  S. 
Then,  2/a  540°  -  2  5. 

.-.#'==  270°  -5. 
**  —  «'  =  90°-(S-ii). 


.  .sin*a-\     sin(i80o-5)sin(180o-C) 
or,  by  Sees.  42  and  29,     

/—  cos  S  cos (S  —  A )  _ .  __ 

sin£a  =  ^ .     p    >   „ l-  [17] 

1  *         sm  £  sm  C 


~.     .,     ,         .  /—  cos  S  cos  (5  —  i?) 

Similarly,   sinU  =  \ : — - — r^— -• 

J'  2  *         Sin  A  sin  C 

/—  cos  S  cos  (£  —  C) 
sin  £  c  =  \ : —      .v      *■ 

"  *         sini  sm  B 

2.   To  express  cos  \  a  m  terms  of  functions  of  A,  B,  C. 
Proceeding  as  above, 

\a  =  W°  -\A\    


.,   _  sin  (s'  -  V)  sin  (s'  —  c') 

cos$a  =  sm  \A<  =  \/ * .     /,    •      , L 

1  r  sm  b  sm  c' 

/sin  [90°-(S-,B)]  sin  [90°-(S-C)] 
"  "V       sin  (180°  -  B)  sin  (180°  -  C) 


OBLIQUE   SPHERICAL    TRIANGLES  137 


^  cos  (S  —  B)  cos  (S  —  C)  . 

or  cos  ^  a  =  \ v — : — '    .    ' -•  [18] 

2  *  sm^sinC  L     J 


Similarly,  cos  \  b  =  \ 


sin  A  sin  C 


/cos  (5  —  4)cos(£  —  5) 
cos  £  c  =  \ > — : — ;    .    VD L- 

2  *  sm  ^4  sin  i? 

3.   To  express  tan  ^  a  i/i  terms  of  functions  of  A,  B,  C. 

From  formulas  [17]  and  [18],  we  have,  by  division  and 
reduction, 


cos  (S  —  B)  cos  (S  —  C) 
Similarly,  tan^o 


—  cos  »S  cos  (>S  —  .4) 

tnn^a=\ — — — a-r- —  [19] 

2  *  cos  (S  —  jB)  cos  (5  —  C) 

j    —  cos  5  cos  (S  —  i?) 
~  ^cos  (S  -  4)  cos  (S  -  C) ' 

j    —  cos  S  cos  (S  —  B) 

tan  I  c  =  \ — — — ^— — — *—  • 

*cos  (5  —  4)  cos  (S  —  £) 


97.    Gauss's  Equations.     The  following  equations  are  known 
as  Gauss's  Equations : 

[1G.]  SmK^+^)=COSc^;&)cos^C. 

[2G.]      *'Ki-^ll'^;>Utg. 

[80.]  coSK^+^)=COSc^tft)sin^- 

[4G.]  eosH^-^-^t^sin^. 

To  derive  the  first  of  these  equations  we  proceed  as  follows  : 
We  have,  by  formula  (9), 

sin  $(A  4-  B)  =  sin  £  A  cos  \  B  -j-  cos  J  ^4  sin  £  5. 


138  SPHERICAL   TRIGONOMETRY 

For  sin  ^  A,  cos  \  B,   substitute  their  values  as  given  in 
Sees.  93  and  94,  and  we  have 


^  /sin(s  —  6) sin  (s  —  c)      sms  sm(s-b) 
l.A  cos^.g=\/       v    .     ; ;    .    v '  X ; * l 

*  ■  '  sin  b  sm  c  sin  a  sin  c 

_  sin  (s  —  b)         /sin  s  sin  (s  —  c) 

—  I  —  X   \  r  !      7 

*      sm  a  sm  6 
X  cos  }  C.  By  Sec.  94. 


sin  c  *      sin  a  sin  b 

_  sin  (s  —  b) 
sin  c 
Similarly, 

sin  (s  —  a) 


cos  At  A  sin  IB  =  -^ *■  X  cos  I  C. 

2  2  sine  2 

.     .  , :    .    ^      sin  (s  —  b)+  sin  (s  —  a)         ,  i_ 
.-.  siniM  +5)= * S2- L zcos4C. 

■ v  7  sm  c  * 

But,  by  formula  (15), 
sin  (s  —  b)-\-  sin  (s  —  a)  =  2  sin  \(2s  —  a  —  b)  cos  £  (a  —  b) 

=  2  sin  £  c  cos  £  (a  —  b), 
since  2s-a-5  =  c. 

Also,  by  formula  (20), 

sin  c  =  2  sin  £  c  cos  £  c. 
.     .,.,„.       2  sin  I  c  cos  i  (a  —  b)  ■ 

•'•  Sm  ^  +  *>  =       2sin^cos^       C°S  *  C' 

,int(^+JB)  =  e0fl^at7>)co»tC.  [16.] 

Proceeding  in  the  same  way,  we  find 

sin  (s  —  b)  —  sin  (s  —  a)  „ 

sm  J  (4  -  B)  = * }- * z  cos  \  C 


sm  c 

2  cos  ±(2s  —  a  —  b)  sin  ^  (a  —  5) 
2  sin  £  c  cos  £  c 


or         sin  i(^  -  B)  =  S'm^~b)  cos  |  C.  [2  G.] 

To  derive  formula  [3  G.],  we  have 

cos  ^  (A  -f  £)  =  cos  \  A  cos  ^  B  —  sin  £  J.  sin  \  B. 


OBLIQUE   SPHERICAL   TRIANGLES 


139 


As  before, __^_ 

'             .  "       '     /sin  s  sin  (5  —  a)       sin  s  sin  (5  —  ft) 
COS  l  ^4  cos  I B  =  -V/ : — -A '  X : ^ L 


sin  ft  sin  e 


sin  a  sine 


ins         /sin  (.9  —  a)  sin  (s  —  b) 
in  a       \  sin  &  sin  a 

X  sin  ^  C. 


sin 
sine 
sin  s 


sine 


Similarly, 

sin  ^  .4  sin  ^  i? 


sin  (s  —  c)    . 


sin  c 


,  ,  .    .    _.       sin  5  —  sin  (s  —  c)    . 

'.  cos  ii  +  £)  = : — * L  sin  i  C 

v  y  sm  e  2 


_  2  cos  |-  (2  s  —  c)  sin  \  c 

sin  c 
_  2  cos  ^  (&  4-  b)  sin  ^  e 
2  sin  ^c  cos  £c 


sin^C 
sin  J  C, 


or         cos  J-  (X  +  J5) 


cos  $■  (<&  4-  ft) 


sin  i-  C. 


[3G.] 


cos  £c 

The  derivation  of  formula  [4  G.]  is  left  as  an  exercise. 
98.    Napier's  Analogies.     The  following  equations  are  known 
as  Napier1  s  Analogies. 


•[1  N.] 
[2N.] 
[3N.] 

[4K] 


tan  I  (A  +  B) 


cos  $(a  —  b) 
cos  J  (a  +  b) 


cot  i  C. 


J      sin  £  (a  4- ft)        * 

.  *       ,'    v      cos  i  (4  —  B)  j 

tan  ia  +  J= ff- { tan  i  c. 

•  y      cos  £  (4  4-  £) 

-        .  sin  i(.4  —  J5)^ 

tan  £  (a  -  ft)  =    .     *;,  'tan  £  c. 

v  /      sini(i+J6)         2 


Formula  [1  N.]  is  derived  from  formulas  [1  G.]  and  [3  G.] 
by  division  and  reduction ;  [2K]  similarly  from  [2  G.]  and 
[4  G.]  ;  [3  K]  from  [4  G.]  and  [3  G.]  ;  [4  K]  from  [2  G.] 
and  [1  G.]. 

Let  the  student  perform  the  operations  indicated. 


140  SPHERICAL   TRIGONOMETRY 


SOLUTION   OF   OBLIQUE    SPHERICAL   TRIANGLES 

99.  If  three  parts  of  a  spherical  triangle  be  given,  the 
remaining  parts  can  be  found  by  the  use  of  formulas  derived 
in  this  chapter.     There  are  six  cases,  as  follows  : 

1.  Given  the  three  sides. 

2.  Given  the  three  angles. 

3.  Given  two  sides  and  their  included  angle. 

4.  Given  two  angles  and  their  included  side. 

5.  Given  two  sides  and  the  angle  opposite  one  of  them. 

6.  Given  two  angles  and  the  side  opposite  one  of  them. 

The  formulas  for  the  tangents  of  the  half  angles  (Sec.  94), 
the  law  of  sines,  and  Gauss's  Equations  or  Napier's  Analogies 
will  be  found  sufficient  for  all  the  cases.  The  results  should 
be  checked  by  formulas  involving  the  required  parts,  not  used 
in  the  solution.  The  method  of  solving  will  be  illustrated  by 
examples. 

100.  Case  1.    Given  the  three  sides. 


Let  a  =  72°  16',  b  =  80°  44',  c  =  41°  18',  to  find  A,  B,  C. 
By  formula  [15],  tan  \A 


/sin  (s  —  b)  sin  (s  —  c) 


sin  s  sin  (s  —  a) 
s=l(a  +  b  +  c)  =  Q7°  9'. 
s  -  a  =  24°  53'. 
s  -  b  =  16°  25'. 
s  -  c  =  55°  51'.     • 


log  sin  16°  25'  =    9.45120  -  10 

log  sin  55°  51'  =    9.91781  -  10 

19.36901  -  20 

9.62066  -  10 

2)19.74835-20 

9.87418  -  10 

\A  =  36°  48'  47". 
A  =  73°  37'  34". 


log  sin  97°  9'    =9.99661-10 

log  sin  24°  53'  =  9.62405  -  10 

9.62066  -  10 


OBLIQUE    SPHERICAL   TRIANGLES 


141 


Similarly,  ta 

log  sin  24°  53'  =    9.62405 

log  sin  55°  51'  =    9.91781 

19.54186 

9.44781 


\      si 


a)  sin  (s  —  c) 


10 
JLO 
20 
10 


sin  s.sin  (s  —  b) 
log  sin  97°  9'    =  9.99661  -  10 
log  sin  16°  25'  =  9.45120  -  10 
9.44781  -  19 


2)20.09405-20 
log  tan  \B  =  10.04703  -  10 
15  =  48°  5' 48". 
B  =  96°  11'  36". 


Similarly, 

log  sin  24°  53' 
log  sin  16°  25' 


->> 


9.62405  -  10 
9.45120  -  10 


19.07525 
9.91442 


20 
10 


sin  (s  —  a)  sin  (s  —  b) 

sin  s  sin  (s  —  c) 
log  sin  97°  9'  =  9 
log  sin  55°  51'  =  9. 


99661  -  10 
91781  -  10 


9.91442  -  10 


•2)1^16083-20 


logtan£C  =    9.58042-10 

}  C  =  20°  50'  5". 
"  C  =  41°  40'  10". 

Check.     From  the  law  of  sines, 

sin  A      sin  B      sin  C 

sin  a       sin  b       sin  c  ' 
from  which 

log  sin  A  —  log  sin  a  =  log  sin  B  —  log  sin  b  =  log  sin  C 

—  log  sin  c. 

log  sin  ^1  =  9.98202-10 
log  sin  a  =9.97886^10 
0.00316 

log  sin  B  =  9.99746-10 
log  sin  6  =9.99429-10 
0.00317 

log  sin  C 
log  sin  c 

=  9.82271- 

=  9.81955- 

0.00316 

-10 
10 

A,  Bj  C  might  also  b( 

j  calculated  by  the  formu 

las  of  Sec. 

95. 

101.    Case  2.    Given  the  three  angles. 

Given  A  =  96°  45',  B  =  108°  30',  C  =  116°  15',  to  find  a,  b,  c. 

From  the  values  of  A,  B,  C  find  a',  6',  c',  the  sides  of  the  polar  triangle 
A'B'C,  by  Sec.  83,  proposition  6.  Solve  the  triangle  A'B'C  by  Case  1, 
finding  A',  B\  C".     From  the  values  of  A",  B',  C"  find  a,  6,  c. 

^4ns.  a  =  88°  27'  49",  b  =  107°  19'  52",  c  =  115°  28'  13". 


Check.    Apply  the  law  of  sines,  as  in  Case  1. 
solved  by  the  formulas  of  Sec.  96. 


This  case  could  also  be 


142  SPHERICAL    TRIGONOMETRY 

102.    Case  3.    Given  two  sides  and  their  included  angle. 

Let  a  =  95°  42',  b  =  72°  22',  C  =  62°  34',  to  find  A,  B,  c. 

By  substituting  the  values  of  a,  6,  C  in  formulas  [1  N.]  and  [2N.],  we 
can  find  A  and  B.     Thus, 

tan  i(A  +  B)  =  cosll°40' .  cot  31o  iTm 
cos  84°  2' 

log  cos  11°  40'  =    9.99093  -  10 

log  cot  31°  17'  =  10.21637  -  10 

20.20730  -  20 

log  cos  84°  2'  =    9.01682  -  10 

log  tan  ±(A  +  B)  =  11.19048  -  10 

$(A  +  B)  =  86°  18'  W, 

tsin  110  ac\' 

and  tan  ±(A-B)=  •  cot  31°  17'. 

v  '       sin  84°  2' 

log  sin  11°  40'  =    9.30582  -  10 

log  cot  31°  17'  =  10.21637  -  10 

19.52219-20 

log  sin  84°  2'  =    9.99764  -  10 

log  tan  £  (A  -  B)  =    9.52455  -  10 

i(A-B)=    18°  30'    4". 

But  i(A  +  B)  =    86°  18'  35". 

.-.  A  =  104°  48'  39". 

B=    67°  48' 31". 

_    _    _  ■  «  sin  c       sin  b 

To  find  c,  we  nave = 

sin  C     sin  B 

log  sin  72°  22'  =    9.97910  -  10- 

log  sin  62°  34'  =    9.94819  -  10 

19.92729  -  20 

log  sin  67°  48'  31"  =    9.96655  -  10 

log  sin  c  =    9.96074-10 

c  =  66°  0'  10". 

The  supplement  of  this  cannot  be  used  as  a  value  of  c,  for  C<B. 
.-.  c  <  b. 

c  might  also  have  been  found  from  formulas  [3N.],  [4  N.],  or  any  one 
of  Gauss's  equations. 

Check.    By  [1  G.],  sin  l(A  +  B)  =  cos^(a~b)  cos  $  C. 

cos  -£-  c 


OBLIQUE    SPHERICAL   TRIANGLES 


143 


103.  Case  4.     Given  two  angles  and  their  included  side. 

Given  A  =  116°,  B  =  84°,  c  =  72°,  to  find  a,  b,  C. 

Solve  the  polar  triangle  by  Case  3.     (See  Sec.  83,  proposition  6.) 

Ans.  a  =  115°  27'  47",  b  =  92°  27'  47",  C  =  71°  12'  45". 

104.  Case  5.     Given  two  sides  and  the  angle  opposite  one  of 
them. 

Let  a  =  41°  6',  b  =  48°  22',  ^  =  54°  17',  to  find  B,  C,  c. 
To  find  B,  we  have,  from  the  law  of  sines, 
sin  B      sin  A 


sin& 
log  sin  48°  22' 
log  sin  54°  17' 


sin  a 

9.87356  -  10 
9.90951  -  10 


19.78307  -  20 

log  sin  41°  6'  =    9.81781  -  10 

log  8^5  =    9.96526-10 

.-.  Bx  =  67°  23'  12",  B2  =  112°  36'  48". 

Since  b  >  a,  we  must  have  B  >  A  ;  but  both  the  values  of  B  found  above 
satisfy  this  condition,  and  we  have  two  solutions. 
Case  5  is  accordingly  an  ambiguous  case.     (See 
Fig.  63.) 

To  find  C,  we  have,  from  formula  [IN.], 
cos  |  (b  +  a) 


cotiC 


tani(£  +  A). 


cos  £  (b  —  a) 
1.  Using  the  first  value  of  B,  we  have 

i(B1  +  ^)  =  60°50'6". 
Also,  i  (6  +  a)  =  44°  44'. 

£  (&-«)  =  3°  38'. 

log  cos  44°  44'  =    9.85150  -  10 

log  tan  60°  50'  6"  =  10.25330  -  10 

20.10480-20 

log  cos  3°  38'  =    9.99913  -  10 


To  find  the  value  of  ci, 


logcoHCi  =  10.10567  - 
£  C\  =  38°  5'  51". 
Ci  =  76°  11'  42' 
sin  C\       sin  a 


10 


sin  C      sin  A 


144  SPHERICAL   TRIGONOMETRY 

log  sin  41°  6'  =    9.81781  -  10 

log  sin  76°  11'  42"  =    9.98727  -  10 

19.80508  -  20 

log  sin  54°  17'  =    9.90951  -  10 

logsinci=:    9.89557-10 

d  =  54°  2'  40". 

2.  Using  the  second  value  of  B,  we  have 

£  (B2  +  A)  =  83°  26'  54". 
Solving  as  above,  log  cos  44°  44'  =    9.85150  —  10 
log  tan  83°  26'  54"  =  10.93987  -  10 
20.79137  -  20 
log  cos  3°  38'  =    9.99913  -  10 
log  cot  i  C2  =  10.79224  -  10 
i  C2  =  9°  9'  56". 
C2  =  18°  19'  52". 
And  to  find  the  value  of  c2, 

sin  c2  _  sin  a 
sin  C2      sin  A 
log  sin  41°  6'  =    9.81781  -  10 
log  sin  18°  19'  52"  =    9.49763  -  10 
19.31544  -  20 
log  sin  54°  17'  =    9.90951  -  10 
logsinc2  =    9.40593-10 
c2  =  14°  45'  11". 
Hence,  we  have  for  the  two  solutions : 

1.  For  the  triangle  ABiC  (Fig.  62), 

Bx  =  67°  23'  12",     Ci  =  76°  11'  42",  d  =  54°  2'  40". 

2.  For  the  triangle  AB2C, 

B2  =  112°  36'  48",  C2  =  18°  19'  52",  c2  =  14°  45'  11". 
Check.     By  formula  [1  G.],  sin  \  {A  +  B)  cos  \  c  —  cos  \  (a  -  b)  cos  i  C. 

If,  in  any  example,  after  substituting  the  values  of  a,  6,  A  in  the 

sin5      sini      ,     T)i  L    .  ,,       ,        .        ,    _ 

equation  = ,  sin  B  turns  out  to  be  greater  than  1,  or  log  sin  B 

sin  b       sin  a 
positive,  the  solution  is  impossible.    If  we  find  sin  B  =  1,  or  log  sin  B  =  0, 
then  B  =  90°,  and  the  solution  can  be  completed  by  the  method  explained 
for  right  triangles. 

If  we  find  sin  B  less  than  1,  or  log  sin  B  negative,  there  will  be  two 
corresponding  values  of  J5,  —  namely,  the  angle  found  in  the  table  and  its 


OBLIQUE   SPHERICAL   TRIANGLES  145 

supplement,  —  and  both  must  be  accepted,  giving  two  solutions,  unless 
one  of  them  should  be  incompatible  with  the  condition  that  the  greater 
angle  lies  opposite  the  greater  side.  Such  a  value  of  B  would  have  to 
be  rejected,  giving  one  solution. 

105.    Case  6.    Given  two  angles  and  the  side  opposite  one  of 
them.     This  is  also  an  ambiguous  case. 

Given  A  =  121°,  B  =  108°,  a  =  130°,  to  find  b,  c,  C. 

By  Case  5,  solve  the  polar  triangle  A'B'C,  finding  the  parts  B',  C",  c', 
and  determine  6,  c,  C  by  proposition  6,  Sec.  83. 

Ans.  1.  6i  =  121°  47'  37",  cx  =  59°  56'  16",     d  =  75°  34'  10". 
2.  62  =  58°  12'  23",     c2  =  160°  29'  42",   C2  =  158°  3'  40". 


EXERCISES 

Solve  the  following  triangles,  checking  the  work  where  the 
answers  are  not  given  : 

1.  Given  a  =  110°  4',   b  =  74°  32',  c  =  56°  30',  to  find  A, 
B,  C. 

Ans.  A  =  127°  35'  42",  B  =  54°  23'  30",  C  =  44°  42'  14". 

2.  Given  a  =  58°  18',  b  =  62°  40',   c  =  78°  24',   to  find  A, 
B,  C.    Ans.  A  =  60°  8'  54",  £  =  64°  54'  8",  C  =  93°  2'  56". 

3.  Given  A  =  53°  50',  B  =  96°  18',   C  =  64°  20',  to  find  a, 
b,  c.       Ans.  a  =  52°  43'  26",  b  =  78°  26'  10",  c  =  62°  40'  22". 

4.  Given  A  =  80°  40',  B  =  116°  20',  C  =  92°  20',  to  find  a, 
b,  c.       Ans.  a  =  78°  23'  14",  b  =  117°  10'  4",  c  =  97°  19'  4". 

5.  Given  a  =  96°,  &  =  64°,  C  =  108°,  to  find  ^,  5,  c. 

Ans.  A  =  87°  31'  53",  £  =  64°  32'  33",  c  =  108°  47'  15". 

6.  Given  a  =  76°  30',  b  =  48°  10',  C  =  36°,  to  find  A,  B,  c. 

Ans.  A  =  121°  32'  8",  B  =  40°  46'  32",  c  =  42°  6'  43". 

7.  Given  ^  =137°  44',  £  =  153°  48',  c  =  107°  32',  to  find 
a,  6,  C. 

Ans.  a  =  102°  44',  b  =  140°  11'  12",  C  =  138°  53'  24". 


146 


SPHERICAL    TRIGONOMETRY 


8.  Given  A  =  81°  32',  B  =  117°  38',  c  =  103°  48',   to  find 
a,  ft,  C.     Ans.  a  =  75°  58'  44",  ft  =  119°  39'  22",  C  =  98°  5'. 

9.  Given  a  =  100°,  b  =  62°,  A  =  95°,  to  find  B,  C,  c. 

Ans.  B  =  63°  16'  17",  C  =  98°  28'  28",  c  =  102°  6'. 

10.  Given  a  =  59°,  6  =  72°,  A  =  50°,  to  find  B,  C,  c. 

Ans.  1.  Bl  =  58°  12'  23",    Cx  =  120°  3'  40",  Cl  =  104°  25'  40". 
2.  52  =  121°  47'  37",  C2  =  19°  30'  14",  c2  =  21°  56'  15". 

11.  Given  A  =  82°,  B  =  116°,  a  =  86°,  to  find  C,  ft,  c. 

Jrcs.  C  =  79°  49'  20",  b  =  115°  7'  10",  c  =  82°  32'. 

12.  Given  A  =  137°  20',  J5  =  64°,  a  =  149°,  to  find  b,  c,  C. 
Ans.  1.  bx  =  136°  55'  9",  Cl  =  19°  45'  24",    Cl  =  26°  24'  40". 

2.  ft2  =  43°  4'  51",    c2  =  130°  43'  50",  C2  =  94°  19'. 

13.  Given  a  =  51°  40',  b  =  72°  12',  A  =  47°  20',  to  find  B,  C,  c. 

14.  Given  a=76°20',  ft  =  110°  24',  c  =  58°  12',  tofind^^C. 

15.  Given  A  =  53°,  B  =  46°,  C  =  127°,  to  find  a,  b,  c. 

16.  Given  A  =  94°,  B  =  62°,  C  =  54°,  to  find  a,  b,  c. 

17.  Given  a  =  62°  12',  ft  =  95°,  C  =  107°  10',  to  find  A,  B,  c. 

18.  Given  a  =  102°,  ft  =  139°,  C  =  132°,  to  find  A,  B,  c. 

19.  Given  A  =  118°,  B  =  82°,  c  =  103°  12',  to  find  a,  ft,  C. 

20.  Given  A  =  156°,  £  =  138°,  c  =  106°,  to  find  a,  ft,  C. 

106.  Latitude  and  Longitude. 
The  latitude  and  longitude  of 
a  point  on  the  surface  of  the 
earth  (regarded  as  a  sphere)  are 
its  arc  coordinates  with  refer- 
ence to  the  equator  and  a  prime 
meridian,  usually  taken  to  be 
the  meridian  passing  through 
Greenwich.  Let  P  and  P'  (Fig. 
64)  be  the  north  and  south  poles 
respectively,  PEP'  the  prime 
meridian,   ECDQ   the  equator, 


OBLIQUE    SPHERICAL   TRIANGLES  147 

and  A  and  B  any  two  points  whose  latitudes  and  longitudes 
are  given.  The  longitude-difference  of  the  two  points  is  the 
are  CD,  which  measures  the  angle  APB.  The  arcs  CA  and 
DB  are  the  latitudes  of  A  and  B  respectively,  and  determine 
the  arcs  AP,  BP. 

In  the  spherical  triangle  APB,  therefore,  we  have  given  two 
sides  and  their  included  angle.  The  arc  AB  can  be  found  in 
degrees  by  Case  3,  Sec.  102. 

To  find  AB  in  miles,  the  radius  of  the  earth  may  be  taken 
as  3956  miles,  from  which  the  length  of  an  arc  of  one  degree 
is  found  to  be  69.05  miles  approximately. 

MISCELLANEOUS  EXERCISES 

1.  Find  the  distance  from  San  Francisco  (latitude  37°  47'  55"  N., 
longitude  122°  24'  32"  W.)  to  Honolulu  (latitude  21°  18'  N.,  longitude 
157°  52' W.).  Ans.  2395  miles. 

2.  Find  the  distance  from  Paris  (latitude  48°  50'  N.,  longitude 
2°  20'  E.)  to  New  York  (latitude  40°  43'  N.,  longitude  74°  0'  W.). 

Ans.  3624  miles. 

3.  Find  the  distance  from  New  Orleans  (latitude  29°  58'  N. ,  longitude 
90°  3'  W.)  to  Rome  (latitude  41°  54'  N.,  longitude  12°  29'  E.). 

Ans.  5444  miles. 

4.  Find  the  distance  from  San  Francisco  to  Manila  (latitude  14°  36'  N. , 
longitude  120°  58'  E.).  Ans.  5465  miles. 

5.  How  many  miles  does  a  vessel  go  in  sailing  due  west  from  a  point 
in  latitude  37°  N.,  longitude  122°  W.,  to  a  point  in  latitude  37°  N.,  longi- 
tude 150°  E.?  How  much  would  the  distance  be  shortened  by  sailing 
along  the  arc  of  a  great  circle  ?  Ans.  4853  miles  ;  200  miles. 


APPENDIX 


LOGARITHMIC  AND  TRIGONOMETRIC   TABLES; 
EXPLANATION   OF   THEIR   USE 

TABLE   I 

Table  I  gives  the  logarithms  of  numbers  from  1  to  10,000. 
On  page  1,  both  the  characteristic  and  the  mantissa  for 
numbers  from  1  to  100  are  given.  In  the  rest  of  the  table 
only  the  mantissa  is  given';  the  characteristic  is  to  be  deter- 
mined by  Sec.  12. 

1.   To  find  the  logarithm  of  a  given  integral  number. 

(a)  Let  the  number  be  less  than  100. 

Look  for  the  number  in  column  "N,"  page  1 ;   opposite  to  it  in  the 
column  "log"  will  be  found  its  logarithm.     Thus, 
log  38  =  1.57978. 

(b)  Let  the  number  consist  of  three  figures. 

Look  for  the  number  in  column  "N,"  and  find  the  mantissa  on  the 
same  horizontal  line  under  column  "0."  By  Sec.  12,  the  characteristic 
is  2.     Thus,  we  find  (page  6) 

log  328  =  2.51587. 

(c)  Let  the  number  consist  of  four  figures. 

Find  the  first  three  figures  in  column  "N";  on  the  same  horizontal 
line,  in  the  column  headed  by  the  last  figure,  find  the  mantissa.     By 
Sec.  12,  the  characteristic  is  3.     Thus,  we  find  (page  15) 
log  7683  =  3.88553. 

(d)  Let  the  number  consist  of  more  than  four  figures.  The 
logarithm  will  be  found  by  the  process  of  interpolation,  which 
will  be  illustrated  by  an  example. 

148 


APPENDIX  149 

.» 

Kequired  the  logarithm  of  58736. 

Place  a  decimal  point  after  the  fourth  figure,  giving  5873.6.  By  Sec.  12, 
this  does  not  affect  the  mantissa.  The  number  5873.6  lies  between  5873 
and  5874. 

By  (c),  the  mantissa  for  5874  is  .76893 

and  for  5873  it  is  .76886 

Difference  for  1  unit  (called  tabular  difference),    .00007 

That  is,  when  the  number  5873  is  increased  by  1  unit,  the  corresponding 
mantissa  is  increased  by  7  hundred-thousandths,  or  7  points,  as  we 
may  conveniently  call  this  increase.  It  is  assumed  that  if  a  number  is 
increased  by  a  small  amount,  the  increase  in  the  logarithm  is  propor- 
tional to  the  increase  in  the  number.  This  assumption  is  not  mathe- 
matically correct,  but  the  error  resulting  from  it  is  so  small  that  it  may 
be  neglected  in  practical  computations. 

Now,  5873.6  exceeds  5873  by  T%  of  a  unit. 

Then,  by  the  above  assumption,  since  the  addition  of  a  whole  unit  to 
5873  increases  the  logarithm  by  7  points,  the  addition  of  T6ff  of  a  unit 
increases  the  logarithm  by  -^  x  7  =4.2,  or  4  points,  nearly. 

Then,  for  5873  the  mantissa  is  .76886 

Add  the  correction,  4 

and  we  have  as  the  mantissa  for  5873.6,      .76890 

This  is,  as  stated,  also  the  mantissa  for  58736.     The  characteristic  is  4. 

Hence, 

log  58736  =  4.76890. 

As  another  example,  let  it  be  required  to  find  the  logarithm 
of  367356. 

As  above,  the  mantissa  is  the  same  as  for  3673.26. 
The  mantissa  for  3674  is        .56514 
and  for  3673  it  is  .56502 

Difference  for  1  unit,  12  points. 

Hence,  the  correction  is  .56  x  12  =  6.72,  or  7  points,  nearly. 

Adding  this  correction,  we  have  for  the  required  mantissa,  .56509. 
The  characteristic  is  5.     Hence, 

log  367356  =  5.56509. 

Note.  When  the  fractional  part  of  a  "point"  is  less  than  .5,  it  is  to  be 
rejected,  as  in  the  first  example  in  (d) ;  while  if  it  is  .5  or  more,  it  is  to  be 
counted  as  a  whole  point,  as  in  the  second  example. 


150  APPENDIX 

The  same  thing  is  done  in  computing  the  mantissas  given  in  the  tables. 
These  mantissas,  therefore,  are  only  approximations,  correct  within  1  hun- 
dred thousandth.  When  the  last  figure  of  the  mantissa  is  written  5,  it  means 
that  the  correct  value  is  slightly  less  than  5. 

(e)  So  far  we  have  considered  only  integral  numbers.  Let 
the  number  whose  logarithm  is  sought  be  a  decimal  fraction, 
or  a  whole  number  plus  a  decimal.  The  characteristic  is  to 
be  determined  by  inspection  in  accordance  with  Sec.  12. .  The 
position  of  the  decimal  point  does  not  affect  the  mantissa. 
Hence,  we  may  in  any  case  omit  the  decimal  and  find  the 
mantissa,  as  in  the  case  already  explained.  Thus,  for  0.28  the 
mantissa  is  the  same  as  for  28  and  is  found  by  (a)  ;  for  0.6834 
it  is  the  same  as  for  6834,  and  is  found  by  (c)  ;  for  3.8757  or 
0.38757  or  0.0038757  it  is  the  same  as  for  38757,  and  is 
found  by  (d). 

For  example,  let  it  be  required  to  find  the  logarithm  of 
0.067374. 

The  mantissa  is  the  same  as  for  67374,  and  is  found  to  be  .82849.  By 
Sec.  12  the  characteristic  is  8  —  10.     Hence, 

log  0.067374  =  8.82849  -  10. 

EXERCISES 
Find  the  logarithms  of  the  following  numbers  : 
\i  \    1.    632.  2.    6320.  3.    6327. 

4.  6327.4.  Arts.  3.80123. 

5.  83545.  Ans.  4.92192. 

6.  297364.  Ans.  5.47329. 
N  V7.    12.36. 

3.  8.1267.  Ans.  0.90992. 

9.  0.31. 

10.  0.0052683.  Ans.  7.72167  -  10. 

11.  0.43687.  Ans.  9.64035-10. 

12.  0.000786. 


APPENDIX  151 

2.    To  find  the  number  corresponding  to  a  given  logarithm. 
The  mantissa  determines  the  figures  of  the  required  number ; 
the  characteristic  determines  the  position  of  the  decimal  point. 

(a)  Let  log  x  =  0.75785. 

The  given  mantissa  is  found  in  the  table  (page  11)  in  column  "  6  "  ;  and 

in  the  same  horizontal  line,  in  column  "N,"  are  found  the  figures  572. 

Hence,  the  figures  of  the  required  number,  without  regard  to  the  position 

of  the  decimal  point,  are  5726.    Placing  the  decimal  point  in  accordance 

with  Sec.  12,  we  have 

x  =  5.726. 

Similarly,  if  log  x  =  2.70561,  x  =  507.7  ; 

and  if  log x  =  8.61574  -  10,  x  =  0.04128. 

(b)  Let  log  x  =  7.45823  -  10. 

The  next  greater  mantissa  is   .45834 

The  next  less  mantissa  is  .45818    which  corresponds  to  2872. 

Difference  for  1  unit,  16  points. 

The  given  mantissa  exceeds  .45818  by  5  points.     Hence,  2872  must  be 
increased  by  T5,j  of  a  unit,  giving  2872T\,  or  2872.3,  nearly.     Placing  the 
decimal  point  in  accordance  with  Sec.  12,  we  have 
x  =  0.0028723. 

EXERCISES 

Find  the  numbers  corresponding  to  the  following  logarithms.     When 
interpolation  is  necessary,  find  the  answer  as  far  as  Jive  significant  figures. 

1.  1.26858.  Ans.  18.56. 

2.  9.26858  -  10.  Ans.  0.1856. 

3.  1.26850.  Ans.  18.5567 

4.  2.88572.  Ans.  768.63. 

5.  0.66396.  Ans.  4.6128. 

6.  7.61817  -  10.  Ans.  0.0041512. 

7.  6.63914  -  10.  Ans.  0.00043565. 

8.  3.46867.  Ans.  2942.2. 

9.  8.31624-10.  Ans.  0.020713. 


152  APPENDIX 

10.  0.23561.  Arts.  1.7203. 

11.  9.59794  -  10.  Ans.  0.39623. 

12.  8.55673  -  10.  Ans.  0.036036. 

TABLE   II 

Logarithmic  Functions  of  Angles 

Table  II  gives  the  logarithms  of  the  sines,  cosines,  tangents, 
and  cotangents  of  angles  from  0°  to  90°  at  intervals  of  1',  except 
as  indicated.  If  the  angle  is  less  than  45°,  the  number  of 
degrees  it  contains  is  to  be  found  at  the  top  of  the  page,  the 
number  of  minutes  in  the  column  for  minutes  on  the  left- 
hand  side  of  the  page,  and  the  name  of  the  function  at  the 
top  ;  if  the  angle  is  greater  than  45°,  the  number  of  degrees  is 
to  be  found  at  the  bottom,  the  number  of  minutes  on  the  right- 
hand  column,  and  the  name  of  the  function  at  the  bottom. 
The  characteristic  is  the  number  at  the  top  or  bottom  of  the 
column  diminished  by  10.  The  numbers  at  the  top  and  bottom 
are  usually  the  same  and  apply  to  all  the  mantissas  in  the 
column ;  the  only  exceptions  are  found  on  pages  29  and  48. 
Here,  the  characteristics  at  the  top  of  the  column  apply  to  all 
the  mantissas  above  the  bars  drawn  across  the  column,  and  the 
characteristics  at  the  bottom  to  the  mantissas  below  the  bars. 

1.  To  find  the  logarithm  of  the  sine,  cosine,  tangent,  or 
cotangent  of  a  given  angle. 

The  use  of  the  table  will  be  understood  from  the  following 
examples : 

(a)  To  find  logarithm  of  the  sine  of  24°  27'. 

This  is  written  log  sin  24°  17',  which  is  sometimes  read  logarithmic  sine 
qf  24°  IT. 

On  page  38,  24°  is  found  at  the  top ;  in  the  column  "log sin,"  in  the 
saiue  horizontal  line  with  27',  we  find  the  mantissa  .61689.  The  charac- 
teristic is  9  —  10.     Hence, 

log  sin  24°  27'  =  9.61689  -  10. 


APPENDIX  153 

"(b)  To  find  log  sin'  24°  27'  13". 

As  above,  log  sin  24°  27'  ==  9.61689  —  10 

log  sin  24°  28'  =  9.61717  -10 
Difference  for  1',  or  60",  28  points. 

The  principle  of  proportional  parts  is  assumed  to  hold  here  also  ;  that 
is,  it  is  assumed  that  if  an  angle  be  increased  by  a  small  amount,  the 
change  in  logarithm  of  each  function  of  the  angle  will  be  proportional  to 
the  increase  in  the  angle. 

Then  if  an  increase  of  1',  or  60",  produces  a  change  of  28  points,  13" 
will  produce  a  change  of  if  x  28  =  6  points,  nearly.  Since  the  sine 
increases  as  the  angle  increases,  this  correction  of  6  points  must  be  added 
to  log  sin  24°  27',  giving 

log  sin  24°  27'  13"  =  9.61695  -  10. 

For  sines  and  tangents,  add  the  correction,  for  these  functions  increase 
as  the  angle  increases  ;  for  cosines  and  cotangents,  subtract  the  correc- 
tion, for  these  functions  decrease  as  the  angle  increases. 

(c)  To  find  log  sin  61°  19'  10". 

Here  the  angle  is  greater  than  45°,  and  we  find  61°  on  page  40  at  the 
bottom  ;  and  looking  in  the  right-hand  column  for  minutes,  we  find 
log  sin  61°  19'  =  9.94314  -  10 
log  sin  61°  20'  =  9.94321  -  10 
Difference  for  60",  7  points. 

The  difference  for  10"  is  accordingly  a§  x  7  =  1  point,  nearly. 
.-.  log  sin  61°  19'  10"  =  9.94315  -  10. 

(d)  To  find  log  tan  73°  24'  8". 

log  tan  73°  24'  =  10.52562  -  10 
log  tan  73°  25'  =  10.52608  -  10 
Difference  for  60",  46  points. 

Difference  for  8"  is,  therefore,  ^\  x  46  =  6  points,  nearly.  Adding 
this  correction,  we  have 

log  tan  73°  24'  8"  se  10.52568  -  10. 

(e)  To  find  log  cos  40°  32'  14". 

log  cos  40°  32'  =  9.88083  -  10 
log  cos  40°  33'  =  9.88072  -  10 
Difference  for  60",  11  points. 


154 


APPENDIX 


The  difference  for  14"  is,  therefore,  $-*  x  11  =  3  points,  Nearly.     Sub- 
tracting this  correction  from  log  cos  40°  32',  we  nave         \ 

log  cos  40°  32'  14"  =  9.88080  -  10.  \ 

(f )  To  find  log  cot  16°  40'  12". 

log  cot  16°  40'  =  10.52378  -  10 
log  cot  16°  41'  =  10.52332  -  10 
Difference  for  60",  40  points. 

The  difference  for  12"  is,  therefore,  £$  x  46  =  9  points,  nearly.     Sub- 
tracting this  correction  from  log  cot  16°  40',  we  have 
log  cot  16°  40'  12"  =  10.52369  -  10. 


EXERCISES 
Find  the  values  of  the  following  : 
Y  V  1.  log  sin  32°  37'  21". 

2.  log  tan  36°  23'  14". 

3.  log  cos  23°  12'  28". 

4.  log  sin  63°  14'. 

5.  log  sin  71°  23'  41". 

6.  log  cot  13°  18'  12". 

7.  log  cot  48°  42'  14". 

8.  log  cos  73°  29' 33". 

9.  logsin71°9'  15". 

10.  log  tan  67°  27'  50". 

11.  log  sin  81°  52' 43". 

12.  log  cos  47°  12'  48". 


Ans.  9.73167-10. 

Ans.  9.86742  -  10. 

Ans.  9.96336-10. 

Ans.  9.95078  -  10. 

Ans.  9.97669  -  10. 

Ans.  10.62626  -  10. 

Ans.  10.94369  -  10. 

Ans.  9.45353  -  10. 

Ans.  9.97607  -  10. 

Ans.  10.38200  -  10. 

Ans.  9.99562  -  10. 

Ans.  9.83205  -  10. 


2.    To  find  the  value  of  an  angle  when  one  of  its  logarithmic 
unctions  is  given. 

(a)  Given  log  sin  x  =  9.77473  -  10. 

The  given  mantissa  is  found  in  the  table  (page  44)  under  36°,  in  column 
log  sin  "  opposite  32'. 

.-.  x  =  36°  32'. 


APPENDIX  155 

(b)  Given  log  sin  x  =  9.77478  —  10,  to  find  the  value  of  x. 
The  given  mantissa  is  not  found  in  the  table,  hut  it  lies  between  .77473 

and  .77490.  Hence,  the  required  angle  lies  between  36°  3T  and  36°  33% 
the  angles  corresponding  to  these  two  ™nt»M 

The  next  greater  mantissa  is  .77490 

The  next  less  mantissa  is         .77473    which  corresponds  to  36°  S^. 

Difference  for  1',  17    points. 

The  given  mantissa  exceeds  the  next  less  mantig»  by  5  points ;  and 
as  the  addition  of  17  points  to  the  next  less  mantissa  increases  the  corre- 
sponding angle  by  1',  the  addition  of  5  points  increases  it  by  ^  x(*=  _  ~   . 

nearly. 

.-.  x  =  36°  3?  18". 

(c)  Given  log  tan  x  =  10.26171  —  10,  to  find  the  value  of  x. 

On  page  40,  in  column  marked  *•  log  tan  "  at  the  bottom  of  the  page, 
we  find 

the  next  less  mantissa  to  be  .26163    which  corresponds  to  61°  18% 

and  the  next  greater  mantissa,       .26193 
Difference  for  1',  30    points. 

The  given  mantissa  exceeds  the  next  less  mantissa  by  8  points.    Hence, 
the  correction  is  ^  x  60  =  16".     Adding  this,  we  have 
x  =  61°  18'  16~. 

(d)  Given  log  cos  x  =  9.94809  —  10,  to  find  the  value  of  x. 

The  next  greater  mantissa  is  .94813  V* 

The  next  less  mantissa  is         .94806  5  which  corresponds  to  27°  28*. 
Difference  for  1',  7    points. 

The  given  mantissa  exceeds  the  next  less  mantissa  by  3  points.  Hence, 
tne  correction  is  4  x  60  =  26",  nearly.     Subtracting  this  from  27°  28% 

we  have 

x  =  273  27'  34". 

(e)  Given  log  cot  x  =  10.13280  —  10,  to  find  the  value  of  x. 
The  next  greater  mantissa  is  .13291 

The  next  less  mantissa  is         .13264    which  corresponds  to  36°  23'. 

Difference  for  1',  27    points. 

The  given  mantissa  exceeds  the  next  less  mantissa  by  16  points. 

Hence,  the  correction  is  \i  x  60  =  36",  nearly.     Subtracting  this  from 

- 

x  =  3e°2r24". 


156  APPENDIX 


EXERCISES 


Determine  the  value  of  x  in  each  of  the  following  equations  : 

\  \  1.  log  sin  x  =  9.75515  -  10.  Ans.  34°  41'  3". 

2.  logtanx  =  9.71257  -  10.  Ans.  27°  17'  21". 

/  ^3.  logtanx  =  10.79774  -  10.  Ans.  80°  56'  53". 

4.  log  sin  x  =  9.97527  -  10.  Ans.  70°  50'  48". 

5.  log  sin  x  =  9.92939  -  10.  Ans.  58°  12'  23". 

6.  log  cos  x  =  9.82678  -10.  Ans.  47°  50' 56". 

7.  log  cos  x  =  9.82431  -  10.  Ans.  48°  8'  32". 

8.  log  cos  x  =  9.58503  -  10.  Ans.  67°  22'  48". 

9.  log  cot  x  =  10.33857  -  10.  Ans.  24°  38'  9". 
J  J 10.  log  cot  x  =  10.76478  -  10.  Ans.  9°  45'  9". 

11.  log  cot  x  =  9.98060  -  10.  Ans.  46°  16'  46". 

12.  log  tan  x  =  10.24009  -  10.  Ans.  60°  5'  14". 

TABLE   III 

For  certain  functions  of  angles  near  0°  or  90°,  the  assump- 
tion involved  in  the  principle  of  proportional  parts  leads  to 
errors  too  great  to  be  neglected  ;  and  for  more  accurate  results 
Table  III  should  be  used.  By  this  table  we  can  determine 
more  accurately  than  by  Table  II, 

1.  log  sin,  log  tan,  log  cot,  for  angles  between  0°  and  2°  ; 

2.  log  cos,  log  tan,  log- cot,  for  angles  between  88°  and  90° ; 

3.  the  angle  from  the  logarithmic  functions  in  the  corre- 
sponding cases. 

The  numbers  S  and  T  are  calculated  to  satisfy  the  relations 

log  sin  x  =  log  x  +  S, 
log  tan  x  =  log  x  +  T, 

x  being  expressed  in  seconds ;  it  being  understood  also  that 
—  10  is  to  be  written  after  each  of  the  numbers  in  the  columns 
"S"  and  "T." 


APPENDIX  157 


EXAMPLES 

1.  To  find  log  sin  1°  13'  26". 

1°  18'  26"  =  4406". 
By  referring  to  the  column  headed  "  x",' '  the  number  4406  is  found  to 
lie  between  4190  and  4840,  and  the  corresponding  number  in  the  column 
"£"  is  4.68554.     We  have,  therefore, 

log  4406  =  3.64404 

S  =  4.68554  -  10 
.-.  log  sin  1°  13'  26"  =  8.32958  -  10 

2.  To  find  log  tan  1°  33' 14". 

1°  33'  14"  =  5594" 
log  5594  =  3.74772 

T  =  4.68568  -  10 


.-.  log  tan  1°  33'  14"  =  8.43340  -  10 

3.  To  find  log  cot  88°  26' 46". 

The  cotangent  of  an  angle  equals  the  tangent  of  its  complement. 
Hence, 

log  cot  88°  26'  46"  =  log  tan  1°  33'  14"  =  8.43340  -  10,  by  Example  2. 

4.  To  find  log  cos  88°  31'  28". 

log  cos  88°  81'  28"  =  log  sin  1°  28'  32", 
which  may  be  found  as  in  Example  1. 

5.  To  find  log  tan  88°  26'  46". 
tan  88°  26'  46"  =  cot  1°  33'  14' 


tan  1°  33'  14' 
.-.  log  tan  88°  26'  46"  =  log  1  -  log  tan  1°  33'  14" 

=  io  -  10  -  log  tan  1°  33'  14". 
10  -10 

By  Example  2,  log  tan  1°  33'  14"  =    8.43340  -  10 
By  subtraction,  log  tan  88°  26'  46"  =    1 .  56660 

6.    To  find  log  cot  88°  26'  46". 

log  cot  88°  26'  46"  =  log  tan  1°  33'  14"  =  8.43340  -  10. 


158 


APPENDIX 


Determine  x  from  each  of  the  following  equations : 

7.  log  sin  x  =  8.38426  -  10. 

From  the  first  of  the  above  formulas,  we  have 

log  x  =  log  sin  x  —  S. 
log  sin  x-  8.38426  -10 

S  =  4.68553  -  10 
logx  =  3.69873 

x  =  4997".2  =  l°23'17".2. 

8.  Log  tan  x  =  1.63283. 

Let  y  be  the  complement  of  x.     From  the  first  of  the  above  formulas, 
log  y  =  log  tan  y  -  T. 

log  tan  y  =  log  cot  x  =  log  (- — -)  =  10  -  10  -  log  tan  x. 
\  tan  x  / 

10  -10 

log  tan  x  =    1.63283 

log  tan  y=    8.36717  -10 

T=    4.68565-10 

log  y=    3.68152 

y  =  4803".l  =  l°20/3,/.l. 

90°  -  y  =  x  =  88°  39r  56".  9. 

EXERCISES 


Find  the,^alues  of  the  following  : 

2'  18".  Ans.  8.42887  -  10 

3".  Ans.  8.25768  -  10 

3.  log  cot  89°  3'  12".  Ans.  8.21811  -  10 

4.  log  cos  88°  42'  15".  Ans.  8.35439  -  10 

5.  log  tan  88°  37' 25".  Ans.  1.61930 
log  cot  88°  32'  8".  Ans.  8.40565  -  10 

x  from  each  of  the  following  equations  : 

7.  log  sin  x  =  8.46872  -  10.  Ans.  1°  41'  10".3. 

i  >!  8.  log  tan  x  =  8.36423  -  10.  Ans.  1°  19'  30".  7. 

9.  logcotx  =  1.71326-10.  Ans.  1°6'31",3. 

10.  log  cos  x  =  8.42126  —  10.  First  find  y,  the  complement  of  x. 

Ans.  88°  29'  18". 2. 


APPENDIX  159 

TABLE   IV 

This  table  gives,  to  four  places  of  decimals,  the  sines, 
cosines,  tangents,  and  cotangents  of  angles  from  0°  to  90° 
at  intervals  of  1'.  The  principle  of  proportional  parts  is 
assumed,  as  in  the  case  of  the  logarithmic  functions. 

EXAMPLES 

1.  To  find  sin  62°  13' 28". 

sin  62°  13'  =  0.8847 
sin  62°  14'  =  0.8849 
Difference  for  1',  2  points. 

.-.  difference  for  28"  =  ||  x  2  =  1  point,  nearly. 

Adding  this  correction  to  sin  62°  13',  we  have 
sin  62°  13'  28"  =  0.8848. 

2.  Given  cot  A  =  1.8473,  to  find  A. 

1.8473  lies  between  1.8469  which  corresponds  to  28°  26', 
and  1.8482  which  corresponds  to  28°  25'. 

Difference  for  1',  13  points. 

1.8473  exceeds  1.8469  by  4  points.  Therefore,  the  angle  28°  26'  must 
be  diminished  by  T4j  x  60"  =  18",  nearly. 

.-.  A  =  28°  25'  42". 

EXERCISES 

Find  the  values  of  the  following  : 
J  \j\  1.  sin  36°  12'  38".  4.  cot  33°  14'  50". 

2.  cos  42°  18'  15".  5.  cos  48°  37'  18". 

3.  tan  62°  18"  24". 


Determine  A  from  each  of  the  following  equations  : 
Xj  \J  6.  sin^l  =0.4882.  "^Sj  V  8.  cos  ^1  =  0.8423 

7.  tan  4  =  2.2963.  9.  cot  A  =  1.1343 


160  APPENDIX 

110.  From  the  results  of  Exercises  1  to  5  find,  by  use  of  Table  I,  the 
corresponding  logarithmic  functions  of  the  given  angles ;  find  the  same 
by  use  of  Table  II,  and  compare  results.     Thus,  in  Example  1,  we  find 
sin  36°  12'  38"  =  0.5907. 
By  Table  I,  log  0.5707  =  9.77137  -  10. 

.-.  log  sin  36°  12'  38"  =  9.77137  -  10. 
By  Table  II,  we  find 

log  sin  36°  12'  38"  =  9.77141  -  10  ; 
the  discrepancy  of  4  points  being  due  to  the  fact  that  in  Table  IV  the 
results  are  given  only  to  four  places,  while  in  Table  II  they  are  given  to 
five  places. 

11.  Given  log  sin  A  =  9.76574  -  10,  to  find  sin  A  by  Table  I. 

Am.  0.5831. 

12.  Given  log  cos  A  =  9.83264  -  10,  to  find  cos  A.         Ans.  0.6802. 

13.  Given  log  tan  A  =  0.97804,  to  find  tan  A.  Ans.  9.507. 

14.  Given  log  sin  A  =  9.59573  -  10,  to  find  A  without  using  Table  II. 

Ans.  23°  13'. 


The  following  tables  are  reproduced,  by  permission,  from 
electrotypes  from  Wentworth  and  Hill's  Five  Place  Loga- 
rithmic and  Trigonometric  Tables. 


TABLES 


* 


TABLE   I 

• 

THE 

COMMON  OE  BEIGGS  LOGAEITHMS 

OF  THE 

NATURAL  NUMBERS 

From  1  to  10000. 

* 

1-100 

N          log 

N 

log 

N          log 

N 

log 

N 

log 

1    0.00  000 

21 

1.32  222 

41     1.  61  278 

61 

1.  78  533 

81 

1.90  849 

2    0.30103 

22 

1.  34  242 

42     1.  62  325 

62 

1.  79  239 

82 

1.  91 381 

3    0.47  712 

23 

1.  36  173 

43     1.  63  347 

63 

1.  79  934 

83 

1.  91  908 

4    0.60  206 

24 

1.38  021 

44     1.  64  345 

64 

1.  80  618 

84 

1.  92  428 

5    0.69  897 

25 

1.  39  794 

45     1.  65  321 

65 

1.  81  291 

85 

1.  92  942 

6  "o.  77  815 

26 

1.41497 

46    1.66  276 

66 

1.  81 954 

86 

1.  93  4£0 

7    0.84  510 

27 

1.  43  136 

47     1.67  210 

67 

1.  82  607 

87 

1.  93  952 

8    0.90  309 

28 

1.  44  716 

48     1.  68  124 

68 

1.83  251 

88 

1.94  448 

9    0.95  424 

29 

1.  46  240 

49     1.69  020 

69 

1.  83  885 

89 

1.  94  939 

10    1.00  000 

30 

1.  47  712 

50     1.69  897 

70 

1.84  510 

90 

1.  95  424 

11     1.04139 

31 

1.  49  136 

51     1.  70  757 

71 

1.  85  126 

91 

1.  95  904     ! 

12    1.07  918 

32 

1.50  515 

52     1.  71  600 

72 

1.  85  733 

92 

1.96  379 

13     1.11394 

33 

1.51851 

53     1.72  428 

73 

1.  86  332 

93 

1.  96  848     l 

14    1.  14  613 

34 

1.  53  148 

54     1.  73  239 

74 

1.  86  923 

94 

1.97  313 

15     1.17  609 

35 

1.  54  407 

55     1.74  036 

75 

1.  87  506 

95 

1.  97  772     1 

16    1.20  412 

36 

1.  55  630 

56     1.  74  819 

76 

1.  88  081 

96 

1.  98  227     J 

17    1.23  045 

37 

1.  56  820 

57     1.  75  587 

77 

1.  88  649 

97 

1.98  677 

18    1.25  527 

38 

1.57  978 

58     1.76  343 

78 

1.  89  209 

98 

1.  99  123 

19    1.27  875 

39 

1.  59  106 

59    1.  77  085 

79 

1.  89  763 

99 

1.99  564 

20    1.30103 

40 

1.  60  206 

60    1.  77  815 

80 

1.90309 

100 

2.00000 

N          log 

N 

log 

N          log 

N 

log 

N 

log 

1-100 


100-150 


K 

O 

1 

2    3    4 

5 

6 

7 

8 

9 

100 

00000 

00  043 

00  087  00130  00173 

00  217 

00  260 

00  303 

00  346 

00  389 

101 

00  432 

00  475 

00  518  00  561  00  604 

00  647 

00  689 

00  732 

00  775 

00  817 

102 

00  860 

00903 

00  945  00  988  01030 

01072 

01115 

01  157 

01199 

01242 

103 

01284 

01326 

01368  01410  01452 

01494 

01536 

01578 

01620 

01662 

104 

01703 

01745 

01787  01828  01870 

01912 

019S3 

01995 

02  036 

02  078 

105 

02119 

02  160 

02  202  02  243  02  284 

02  325 

02  366 

02  407 

02  449 

02  490 

106 

02  531 

02  572 

02  612  02  653  02  694 

02  735 

02  776 

02  816 

02  857 

02  898 

107 

02  938 

02  979 

03  019  03  060  03  100 

03  141 

03  181 

03  222 

03  262 

03  302 

108 

03  342 

03  383 

03  423  03  463  03  503 

03  543 

03  583 

03  623 

03  663 

03  703 

109 

03  743 

03  782 

03  822  03  862  03  902 

03  941 

03  981 

04  021 

04  060 

04100 

HO 

04139 

04179 

04  218  04  258  04  297 

04  336 

04  376 

04  415 

04  454 

04  493 

111 

04  532 

04  571 

04  610/  04  650  04  689 

04  727 

04  766 

04  805 

04  844 

04  883 

112 

04  922 

04  961 

04  999  05  038  05  077 

05  115 

05  154 

05  192 

05  231 

05  269 

113 

05  308 

05  346 

05  385  05  423  05  461 

05  500 

05  538 

05  576 

05  614 

05  652 

114 

05  690 

05  729 

05  767  05  805  05  843 

05  881 

05  918 

05  956 

05  994 

06032 

115 

06  070 

06108 

06145  06183  06  221 

06  258 

06  296 

06  333 

06  371 

06  408 

116 

06446 

06  483 

06  521  06  558  06  595 

06  633 

06  670 

06  707 

06  744 

06  781 

117 

06  819 

06  856 

06  893  06930  06967 

07  004 

07  041 

07  078 

07115 

07151 

118 

07188 

07  225 

07  262  07  298  07  335 

07  372 

07  408 

07  445 

07  482 

07  518 

119 

07  555 

07  591 

07  628  07  664  07  700 

07  737 

07  773 

07  809 

07  846 

07  882 

120 

07  918 

07  954 

07  990  08  027  08  063 

08  099 

08135 

08171 

08  207 

08  243 

121 

08  279 

0S3i4 

08  350  08  386  08  422 

08  458 

08  493 

08  529 

08  565 

08  600 

122 

08  636 

08  672 

08  707  08  743  08  778 

08  814 

08  849 

08  884 

08  920 

08  955 

123 

08  991 

09  026 

09  061  09  096  09132 

09167 

09  202 

09  237 

09  272 

09  307 

124 

09  342 

09  377 

09  412  09  447  09  482 

09  517 

09  552 

09  587 

09621 

09  656 

125 

09  691 

09  726 

09  760  09  795  09  830 

09  864 

09  899 

09  934 

09968 

10003 

126 

10  037 

10072 

10106  10140  10175 

10  209 

10  243 

10  278 

10  312 

10  346 

127 

10  380 

10  415 

10449  10^83  10  517 

10  551 

10  585 

10  619 

10  653 

10  6S7 

128 

10  721 

10  755 

10  789  10  823.10  857 

10  890 

10  924 

10958 

10992 

11025 

129 

11 059 

11093 

11126  11160  11193 

11227 

11261 

11294. 

11327 

11361 

130 

11394 

"11428 

11461  11494  11528 

11561 

11594 

11628 

11661 

11694 

131 

11727 

11**60 

11793  11826  11860 

11893 

11926 

11959 

11992 

12  024 

132 

12  057 

12  090 

12  123  12  156  12189 

12  222 

12  254 

12  287 

12  320 

12  352 

133 

12  385 

12  418 

12  450  12  483  12  516 

12  548 

12  581 

12  613 

12  646 

12  678 

134 

12  710 

12  743 

12  775  12  808  12  840 

12  872 

12  905 

12  937 

12  969 

13  001 

135 

13  033 

13  066 

13  098  13130  13  162 

13194 

13  226 

13  258 

13  290 

13  322 

136 

13  354 

13  386 

13  418  13  450  13  481 

13  513 

13  545 

13  577 

13  609 

13  640 

137 

13  672 

13  704 

13  735  13  767  13  799 

13  830 

13  862 

13  893 

13  925 

13  956 

138 

13  988 

14  019 

14  051  14  082  14114 

14145 

14176 

14  208 

14  239 

14  270 

139 

14  301 

14  333 

14  364  14  395  14  426 

14  457 

14  489 

14  520 

14  551 

14  582 

140 

14  613 

14  644 

14  675  14  706  14  737 

14  768 

14  799 

14  829 

14  860 

14  891 

141 

14  922 

14  953 

14  983  15  014  15  045 

15  076 

15  106 

15137 

15  168 

15  198 

142 

15  229 

15  259 

15  290  15  320  15  351 

15  381 

15  412 

15  442 

15  473 

15  503 

143 

15  534 

15  564 

15  594  15  625  15  655 

15  685 

15  715 

15  746 

15  776 

15  806 

144 

15  836 

15  866 

15  897  15  927  15  957 

15  987 

16  017 

16  047 

16077 

16107 

145 

16137 

16167 

16197  16  227  16  256 

16  286 

16  316 

16  346 

16  376 

16406 

146 

16435 

16  465 

16495  16  524  16  554 

16  584 

16  613 

16  643 

16673 

16  702 

147 

16  732 

16  761 

16  791  16  820  16  850 

16  879 

16  909 

16  938 

16  967 

16997 

148 

17  026 

17  056 

17  085  17114  17143 

17173 

17  202 

17  231 

17  260 

17  289 

149 

17  319 

17  348 

17  377  17406  17  435 

17464 

17  493 

17  522 

17  551 

17  580 

ISO 

17609 

17  638 

17667  17  696  17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

* 

O 

1 

2    3    4 

5 

6 

7 

8 

9 

100-150 


150-200 

3 

N 

O 

12    3 

4 

5 

6 

7 

8 

9 

150 

17  609 

17  638  17  667  17  696 

17  725 

17  754 

17  7S2 

17  811 

17  840 

17  869 

151 

17S9S 

17  926  17  955  17  934 

IS  013 

18  041 

IS  070 

18  099 

IS  127 

18156 

152 

181S4 

1821*3  IS  241  18  270 

IS  298 

18  327 

IS  355 

IS  334 

18  412 

18  441 

153 

IS  469 

1S49S  IS  526  IS  554 

18  533 

18  611 

IS  639 

IS  667 

IS  696 

18  724 

154 

18  752 

18  7SO  18  808  18  837 

18  865 

18  893 

18  921 

18  949 

18  977 

19  005 

155 

19  033 

19  061  19  089  19117 

19145 

19173 

19  201 

19  229 

19  257 

19  285 

156 

19  312 

19  340  19  368  19  396 

19  424 

19  451 

19  479 

19  507 

19  535 

19  562 

157 

19  590 

19  618  19  645  19  673 

19  700 

19  728 

19  756 

19  783 

19  811 

19  838 

15S 

19S66 

19  893  19  921 .  19  948 

19  976 

20003 

20  030 

20  058 

20  085 

20112 

159 

20140 

20167  20194  20  222 

20  249 

20  276 

20303 

20  330  20  358  203Si 

160 

20  412 

20  439  20  466  20493 

20  520 

20  548 

20  575 

20  602 

20  629 

20  656 

161 

20  6S3 

20  710  20  737  20  763 

20  790 

20  817 

20  844 

20  871 

20  898 

20  925 

162 

20  952 

20  978  21005  21032 

21059 

21085 

21112 

21139 

21165 

21192 

163 

21219 

21245  21272  21299 

21325 

21352 

21378 

21405 

21431 

21458 

164 

21484 

21511  21537  21564 

21590 

21617 

21643 

21669 

21696 

21722 

165 

21748 

21775  21801  21827 

21854 

218S0 

21906 

21932 

21958 

21985 

166 

22  011 

22  037  22  063  22  039 

22  115 

22  141 

22  167 

22  194 

22  220 

22  246 

167 

22  272 

22  298  22  324  22  350 

22  376 

22  401 

22  427 

22  453 

22  479 

22  505 

16S 

22  531 

22  557  22  5S3  22  60S 

22  634 

22  660 

22  6S6 

22  712 

22  737 

22  763 

169 

22  7S9 

22  814  22  840  22  866 

22  891 

22  917 

22  943 

22  968 

22  994 

23  019 

170 

23  045 

23  070  23  096  23  121 

23  147 

23172 

23198 

23  223 

23  249 

23  274 

171 

23  300 

23  325  23  350  23  376 

23  401 

23  426 

23  452 

23  477 

23  502 

23  528 

172 

23  553 

23  578  23  603  23  629 

23  654 

23  679 

23  704 

23  729 

23  754 

23  779 

173 

23  805 

23  830  23  855  23  8S0 

23  905 

23  930 

23  955 

23  980 

24  005 

24  030 

174 

24  055 

24  0S0  24105.  24130 

24155 

24  ISO 

24  204 

24  229 

24  254 

24  279 

175 

24  304 

24  329  24  353  24  37S 

24  403 

24  428 

24  452 

24  477 

24  502 

24  527 

176 

24  551 

24  576  24  601  24  625 

24  650 

24  674 

24  699 

24  724 

24  748 

24  773 

177 

24  797 

24  822  24  846  24  871 

24  895 

24  920 

24  944 

24  969 

24  993 

25  018 

178 

25  042 

25  066  25  091  25  115 

25  139 

25  164 

25  188 

25  212 

25  237 

25  261 

179 

25  2S5 

25  310  25  334  25  358 

25  382 

25  406 

25  431 

25  455 

25  479 

25  503 

180 

25  527 

25  551  25  575  25  600 

25  624 

25  648 

25  672 

25  696 

25  720 

25  744 

181 

25  76S 

25  792  25  816  25  840 

25  864 

25  888 

25  912 

25  935 

25  959 

25  983 

182 

26  007 

26  031  26  055  26  079 

26102 

26126 

26150 

26174 

26198 

26  221 

183 

26  245 

26  269  26  293  26  316 

26340 

26  364 

,26  387 

26  411 

26  435 

26  458 

184 

26482 

26  505  26  529  26  553 

26  576 

26  600 

26  623 

26  647 

26  670 

26  694 

185 

26  717 

26  741  26  764  26  788 

26811 

26  834 

26  858 

26  881 

26  905 

26  928 

186 

26  951 

26  975  26  998  27  021 

27  045 

27  068 

27  091 

27114 

27138 

27  161 

187 

27  1S4 

27  207  27  231  27  254 

27  277 

27  300 

27  323 

27  346 

27  370 

27  393 

188 

27  416 

27  439  27  462  27  485 

27  508 

27  531 

27  554 

27  577 

27  600 

27  623 

189 

27  646 

27  669  27  692  27  735 

27  738 

27  761 

27  784 

27  807 

27  830 

27  852 

190 

27  875 

27  898  27  921  27  944 

27  967 

27  989 

28  012 

28  035 

28  058 

28  081 

191 

28103 

28126  28149  28171 

28194 

2S217 

28  240 

28  262 

28  285 

28  307 

192 

28  330 

28  353  28  375  28  398 

28  421 

28  443 

28  466 

28  488 

2S511 

28  533 

193 

28  556 

28  578  28  601  28  623 

28  646 

28  668 

28  691 

28  713 

2S735 

28  758 

194 

28  780 

2S803  28  825  28  847 

28  870 

28  892 

28  914 

28  937 

28  959 

28  981 

195 

29  003 

29  026  29  048  29  070 

29  092 

29115. 

29137 

29159 

29181 

29  203 

196 

29  226 

29  248  29  270  29  292 

29  314 

29  336 

29  358 

29  380 

29  403 

29  425 

197 

29  447 

29  469  29  491  29  513 

29  535 

29  557 

29  579 

29  601 

29  623 

29  645 

198 

29  667 

29  688  29  710  29  732 

29  754 

29  776 

29  798 

29  820 

29  842 

29  863 

199 

29  885 

29  907  29  929  29  951 

29  973 

29  994 

30  016 

30  038 

30  060 

30  081 

200 

301Q3 
0 

30125  30146  30168 

30190 

30  211 

30  233 

30  255 

30  276 

30  298 

N 

12    3 

4 

5 

6 

7 

8 

9 

150-200 


4 

200-250 

N 

O 

1 

2 

3 

4 

'   5 

6 

7 

8 

9 

200 

30103 

30125 

30146 

30168 

30190 

30  211 

30  233 

30  255 

30  276 

30  298 

201 

30  320 

30  341 

30363 

30  384 

30  406 

30  428 

30  449 

30  471 

30  492 

30  514 

202 

30  535 

30  557 

30  578 

30  600 

30  621 

30  643 

30  664 

30  6S5 

30  707 

30  728 

203 

30  750 

30  771 

30  792 

30  814 

30  835 

30  856 

30  878 

30  899 

30  920 

30  942 

204 

30963 

30984 

31006 

31027 

31048 

31069 

31091 

31112 

31133 

31154 

205 

31175 

31197 

31218 

31239 

31260 

31281 

31302 

31323 

31345 

31366 

206 

31387 

31408 

31429 

31450 

31471 

31492 

31513 

31534 

31555 

31576 

207 

31597 

31618 

31639 

31660 

316S1 

31702 

31723 

31744 

31765 

31785 

208 

31806 

31827 

31848 

31869 

31890 

31911 

31931 

31952 

31973 

31994 

209 

32  015 

32  035 

32  056 

32  077 

32  098 

32118 

32139 

32160 

32181 

32  201 

210 

32  222 

32  243 

32  263 

32  284 

32  305 

32  325 

32  346 

32  366 

32  387 

32  408 

211 

32  428 

32  449 

32  469 

32  490 

32  510 

32  531 

32  552 

32  572 

32  593 

32  613 

212 

32  634 

32  654 

32  675 

32  695 

32  715 

32  736 

32  756 

32  777 

32  797 

32S18 

213 

32  838 

32  858 

32  879 

32  899 

32  919 

32  940 

32  960 

32  9S0 

33  001 

33  021 

214 

33  041 

33  062 

33  082 

33102 

33  122 

33  143 

33163 

33  183 

33  203 

33  224 

215 

33  244 

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33  284 

33  304 

33  325 

33  345 

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33  385 

33  405 

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36  361 

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36  399 

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36  511 

36  530 

232 

36  549 

36  568 

36  586  36605 

36  624 

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36  717 

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36  736 

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36  773 

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36  8S4 

36  903 

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36  922 

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38  202 

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38  399 

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38  721 

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38  792 

38  810 

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38  917 

38  934 

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39  005 

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39  217 

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39  252 

247 

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39  322 

39  340 

39  358 

39  375 

39  393 

39  410 

39  428 

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39  602 

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40  500 

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44  059-  44  075 

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44  13S 

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47  217 

47  232 

47  246 

47  261 

297 

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47  319 

47  334 

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47  392 

47  407 

298 

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300 

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47  82S 

47  842 

301 

47  857 

47  871 

47  885 

47  900 

47  914 

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47  943  47  958 

47  972 

47  986 

302 

48  001 

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48*259 

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48  401 

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307 

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48  982 

309 

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49  010 

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310 

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49  248 

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311 

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49  318 

49  332 

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49  360  49  374 

49  388 

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312 

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49  527 

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313 

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314 

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49  803 

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49  914  49  927 

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316 

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49  9S2 

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50  010 

50  024 

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50  051  50  065 

50  079 

50  092 

317 

50106 

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50  215 

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318 

50  243 

50  256 

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50  325  50  338 

50  352 

50  365 

319 

50  379 

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50  406 

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50  461  50  474 

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50  501 

320 

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50  623 

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50  691 

50  705 

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50  732  50  745 

50  759 

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322 

50  7S6 

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50  826 

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50  893 

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323 

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52  140 

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52  166 

52  179 

52  192  52  205 

52  218 

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333 

52  244 

52  257 

52  270 

52  284 

52  297 

52  310 

52  323  52  336 

52  349 

52  362 

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52  375 

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52  634 

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53  007 

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53  020 

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53  224  53  237 

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53  90S 

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55  169 

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55  206  55  218 

55  230 

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55  955 

55  967 

55  979 

363 

55  991 

56003 

56015 

56  027 

56038 

56050  56062 

56074 

560S6 

56098 

364 

56110 

56122 

56134 

56146 

56158 

56170  56182 

56194 

56  205 

56  217 

365 

56229 

56  241 

56  253 

56  265 

56  277 

56  289  56  301 

56  312 

56  324 

56  336 

366 

56  348 

56360 

56  372 

56  384 

56  396 

56407  56  419 

56  431 

56443 

56455 

367 

56467 

56  47S 

56  490 

56  502 

56  514 

56  526  56  538 

56  549 

56  561 

56  573 

368 

56  585 

56  597 

56  608 

56  620 

56  632 

56  644 '56  656 

56667 

56  679 

56  691 

369 

56  703 

56  714 

56  726 

56  738 

56  750 

56  761  56  773 

56  7S5 

56  797 

56  SOS 

370 

56820 

56  832 

56  844 

56  855 

56  867 

56  879  56  891 

56902 

56914 

56926  | 

371 

56  937 

56949 

56  961 

56  972 

56  984 

56  996  57  008 

57  019 

57  031 

57  043 

372 

57  054 

57  066 

57  07S 

57  0S9 

57101 

57113  57124 

57136 

57  148 

57  159 

373 

57171 

57183 

57  194 

57  206 

57  217 

57  229  57  241 

57  252 

57  264 

57  276 

374 

57  287 

57  299 

57  310 

57  322 

57  334 

57  345  57  357 

57  368 

57  3S0 

57  392 

375 

57403 

57415 

57  426 

57  438 

57449 

57461  57473 

57  4S4 

57  496 

57  507 

376 

57  519 

57  530 

57  542 

57  553 

57  565 

57  576  57  5S8 

57  600 

57  611 

57  623 

377 

57  634 

57  646 

57  657 

57  669 

57  6S0 

57  692  57  703 

57  715 

57  726 

57  738 

378 

57  749 

57  761 

57  772 

57  784 

57  795 

57  807  57  818 

57  830 

57  841 

57  852 

379 

57  864 

57  875 

57  8S7 

57  898 

57910 

57  921  57  933 

57  944 

57  955 

57  967 

380 

57  978 

57990 

58  001 

58  013 

58  024 

58  035  58  047 

58  058 

5S070 

58  081 

381 

58092 

58  104 

5S115 

58127 

58138 

58149  SS161 

58172 

58184 

5S195 

3S2 

58  206 

58  218 

58  229 

58  240 

58  252 

58  263  58  274 

58  286 

58  297 

58  309 

383 

58  320 

5S331 

5S343 

58  354 

58  365 

58  377  5S388 

5S399 

5S410 

58  422 

3S4 

5S433 

58  444 

58  456 

5S467 

58478 

58  490  58  501 

58512 

58  524 

5S53J 

385 

58  546 

5S557 

58  569 

58  5SO 

58  591 

5S602  58  614 

58  625 

58  636 

58  647 

3S6 

58  659 

58  670 

58  6S1 

58  692 

58  704 

58  715  58  726 

58  737 

58  749 

58  760 

387 

58  771 

58  782 

58  794 

58  805 

58  816 

58  827  58  838 

58  850 

58  861 

58  872 

388 

58  8S3 

58  894 

58  906 

5S917 

58  928 

58  939  58  950 

58  961 

58  973 

58  9S4 

389 

58  995 

59  006 

59  017 

59  028 

59040 

59  051  59062 

59  073 

590S4 

59095 

390 

59106 

59118 

59129 

59140 

59151 

59162  59173 

59184 

59195 

59  207 

391 

59  218 

59  229 

59  240 

59  251 

59  262 

59  273  59  284 

59  295 

59  306 

59  318 

392 

59  329 

59  340 

59  351 

59  362 

59  373 

59  3S4  59  395 

59  406 

59  417 

59  428 

393 

59  439 

59  450 

59  461 

59  472 

59  483 

59  494  59  506 

59  517 

59  528 

59  539 

394 

59  550 

59  561 

59  572 

59  583 

59  594 

59  605  59  616 

59  627 

59  638 

59  649 

395 

59  660 

59  671 

59  682 

59  693 

59  704 

59  715  59  726 

59  737 

59  748 

59  759 

396 

59  770 

59  780 

59  791 

59  802 

59  813 

59  824  59  835 

59  846 

59  857 

59  868 

397 

59S79 

59  890 

59  901 

59  912 

59  923 

59  934  59  945 

59  956 

59  966 

59  977 

398 

59988 

59  999 

60  010 

60  021 

60  032 

60043  60  054 

60  065 

60076 

60  086 

399 

60  097 

60108 

60119 

60130 

60141 

60152  60163 

60173 

60184 

60195 

400 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260  60  271 

60  282 

60  293 

60  304 

N 

O 

1 

2 

3 

4 

5    6 

7 

8 

9 

350-400 


8 

400-450 

400 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260 

60  271 

60  282 

60  293 

60  304 

401 

60  314 

60  325 

60  336 

60  347 

60  358 

60  369 

60  379 

60  390 

60  401 

60  412 

402 

60  423 

60  433 

60  444 

60  455 

60  466 

60  477 

60  487 

60  498 

60  509 

60  520 

403 

60  531 

60  541 

60  552 

60  563 

60  574 

60  5S4 

60  595 

60  606 

60  617 

60  627 

404 

60  638 

60  649 

60660 

60  670 

60  681 

60  692 

60  703 

60  713 

60  724  60  735 

405 

60  746 

60  756 

60  767 

60  778 

60  788 

60  799 

60  810 

60  821 

60  831 

60  842 

406 

60  853 

60  863 

60  874 

60  885 

60  895 

60  906 

60  917 

60  927 

60  938 

60  949 

407 

60  959 

60  970 

60  9S1 

60  991 

61002 

61013 

61023 

61034 

61045 

61055 

408 

61066 

61077 

61087 

61098 

61109 

61119 

61  130 

61  140 

61  151 

61  162 

409 

61172 

61183 

61194 

61204 

61215 

61225 

61236 

61247 

61257 

61268 

410 

61278 

61289 

61300 

61310 

61321 

61331 

61342 

61352 

61363 

61374 

411 

61384 

61395 

61405 

61416 

61426 

61437 

61448 

61  458 

61469 

61479 

412 

61490 

61500 

61511 

61521 

61532 

61542 

61553 

61563 

61574 

61584 

413 

61595 

61606 

61616 

61627 

61637 

61648 

61658 

61669 

61,679 

61690 

414 

61700 

61711 

61721 

61731 

61742 

61752 

61763 

61773 

61784 

61794 

415 

61805 

61  815. 

61826 

61836 

61847 

61857 

61868 

61878 

61888 

61899 

416 

61909 

61920 

61930 

61941 

61951 

61962 

61972 

61982 

61993 

62  003 

417 

62  014 

62  024 

62  034 

62  045 

62  055 

62  066 

62  076 

62  086 

62  097 

62107 

418 

62  118 

62  128 

62  138 

62149 

62159 

62170 

62  180 

62  190 

62  201 

62  211 

419 

62  221 

62  232 

62  242 

62  252 

62  263 

62  273 

62  284 

62  294 

62  304 

62  315 

420 

62  325 

62  335 

62  346 

62  356 

62  366 

62  377 

62  387 

62  397 

62  408 

62  418 

421 

62  428 

62  439 

62  449 

62  459 

62  469 

62  480 

62  490 

62  500 

62  511 

62  521 

422 

62  531 

62  542 

62  552 

62  562 

62  572 

62  583 

62  593 

62  603 

62  613 

62  624 

423 

62  634 

62  644 

62  655 

62  665 

62  675 

62  6S5 

62  696 

62  706 

62  716 

62  726 

424 

62  737 

62  747 

62  757 

62  767 

62  778 

62  788 

62  798 

62  808 

62  818 

62  829 

425 

62  839 

62  849 

62  859 

62  870 

62  880 

62  89a 

62  900 

62  910 

62  921 

62  931 

426 

62  941 

62  951 

62  961 

62  972 

62  9S2 

62  992 

63  002 

63  012 

63  022 

63  033 

427 

63  043 

63  053 

63  063 

63  073 

63  083 

63  094 

63  104 

63114 

63  124 

63  134 

428 

63  144  63  155. 

63  165 

63  175 

63  185 

63  195 

63  205 

63  215 

63  225 

63  236 

429 

63  246 

63  256 

63  266 

63  276 

63  286 

63  296 

63  306 

63  317 

63  327 

63  337 

430 

63  347 

63  357 

63  367 

63  377 

63  387 

63  397 

63  407 

63  417 

63  428 

63  433 

431 

63  448 

63  458 

63  468 

63  478 

63  488 

63  498 

63  508 

63  518 

63  528 

63  538 

432 

63  548 

63  558 

63  568 

63  579 

63  589 

63  599 

63  609 

63  619 

63  629 

63  639 

433 

63  649 

63  659 

63  669 

63  679 

63  689 

63  699 

63  709 

63  719 

63  729 

63  739 

434 

63  749 

63  759 

63  769 

63  779 

63  789 

63  799 

63  809 

63  819 

63  829 

63  839 

435 

63  849 

63  859 

63  869 

63  879 

63  889 

63  899 

63  909 

63  919 

63  929 

63  939 

436 

63  949 

63  959 

63  969 

63  979 

63  988 

63  998 

64  008 

64  018 

64  028 

64  038 

437 

64  048 

64  058 

64  068 

64  078 

64  088 

64  098 

64108 

64118 

64128 

64137 

438 

64147 

64157 

64167 

64177 

64187 

64  197 

64  207 

64  217 

64  227 

64  237 

439 

64  246 

64  256 

64  266 

64  276 

64  286 

64  296 

64  306 

64  316 

64  326 

64  335 

440 

64  345 

64  355 

64  365 

64  375 

64  385 

64  395 

64  404 

64  414 

64  424 

64  434 

441 

64  444 

64  454 

64  464 

64  473 

64  4S3 

64  493 

64  503 

64  513 

64  523 

64  532 

442 

64  542 

64  552 

64  562 

64  572 

64  582 

64  591 

64  601 

64  611 

64  621 

64  631 

443 

64  640 

64  650 

64  660 

64  670 

64  680 

64  689 

64  699 

64  709 

64  719 

64  729 

444 

64  738 

64  748 

64  758 

64  768 

64  777 

64  787 

64  797 

64  807 

64  816 

64  826 

445 

64  836 

64  846 

64  856 

64  865 

64  875 

64  885 

64  895 

64  904 

64  914 

64  924 

446 

64  933 

64  943 

64  953 

64  963 

64  972 

64  9S2 

64  992 

65  002 

65  011 

65  021 

447 

65  031 

65  040 

65  050 

65  060 

65  070 

65  079 

65  089 

65  099 

65  108 

65  118 

448 

65128 

65  137 

65  147 

65  157 

65  167 

65  176 

65  186 

65  196 

65  205 

65  215 

449 

65  225 

65  234 

65  244 

65  254 

65  263 

65  273 

65  283 

65  292 

65  302 

65  312 

450 

65  321 

65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400-450 


450-500 

9 

N 

O    1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321  65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408 

451 

65  418  65  427 

65  437 

65  447 

65  456 

65  466 

65  475 

65  4S5 

65  495 

65  504 

452 

65  514  65  523 

65  533 

65  543 

65  552 

65  562 

65  571 

65  581 

65  591 

65  600 

453 

65  610  65  619 

65  629 

65  639 

65  648 

65  658 

65  667 

65  677 

65  686 

65  696 

454 

65  706  •  65  715 

65  725 

65  734 

65  744 

65  753 

65  763 

65  772 

65  782 

65  792 

455 

65  801  65  811 

65  820 

65  830 

65  839 

65  849 

65  858 

65  868 

65  877 

65  887 

456 

65  896  65  906 

65  916 

65  925 

65  935 

65  944 

65  954 

65  963 

65  973 

65  982 

457 

65  992  66  001 

66  011 

66  020 

66  030 

66  039 

66  049 

66  058 

66  06S 

66  077 

458 

66  0S7  66  096 

66106 

66115 

66124 

66  134 

66143 

66153 

66162 

66172 

459 

66181  66191 

66  200 

66  210 

66  219 

66  229 

66  23S 

66  247 

66  257 

66266 

460 

66  276  66  28.5 

66  295 

66  304 

66  314 

66  323 

66  332 

66342 

66  351 

66361 

461 

66  370  66  3S0 

66  3S9 

66  398 

66  408 

66  417 

66  427 

66  436 

66445 

66  455 

462 

66  464  66  474 

66  483 

66  492 

66  502 

66  511 

66  521 

66  530 

66  539 

66  549 

463 

66  558  66  567 

66  577 

66  586 

66  596 

66  605 

66  614 

66  624 

66  633 

66  642 

464 

66  652  66  661 

66  671 

66  680 

66  689 

66  699 

66  708 

66  717 

66  727 

66  736 

465 

66  745  66  755 

66  764 

66  773 

66  783 

66  792 

66  801 

66  811 

66  820 

66  829 

466 

66  839  66  848 

66  857 

66S67 

66  876 

66  885 

66  894 

66  904 

66913 

66  922 

467 

66  932  66941 

66  950 

66  960 

66  969 

66  978 

66987 

66  997 

67  006 

67  015 

468 

67  025  67  034 

67  043 

67  052 

67  062 

67  071 

67  0S0 

67  0S9 

67  099 

67  108 

469 

67117  67127 

67136 

67145 

67154 

67164 

67173 

67  382 

67191 

67  201 

470 

67  210  67  219 

67  228 

67  237 

67  247 

67  256 

67  265 

67  274 

67  284 

67  293 

471 

67  302  67  311 

67  321 

67  330 

67  339 

67  348 

67  357 

67  367 

67  376 

67  385 

472 

67  394  67  403 

67  413 

67  422 

67  431 

67  440 

67  449 

67  459 

67  468 

67  477 

473 

67  486  67  495 

67  504 

67  514 

67  523 

67  532 

67  541 

67  550 

67  560 

67  569 

474 

67  578  67  587 

67  596 

67  605 

67  614 

67  624 

67  633 

67  642 

67651 

67  660 

475 

67  669  67  679 

67688 

67  697 

67  706 

67  715 

67  724 

67  733 

67  742 

67  752 

476 

67  761  67  770 

67  779 

67  788 

67  797 

67  806 

67  815 

67  825 

67  834 

67  843 

477 

67  852  67  861 

67  870 

67  879 

67  888 

67  897 

67  906 

67  916 

67  925 

67  934 

478 

67  943  67  952 

67  961 

67  970 

67  979 

67  988 

67  997 

6S006 

68  015 

68  024 

479 

68  034  68  043 

68  052 

68  061 

68  070 

68  079 

6S0S8 

68097 

68106 

68115 

480 

6S124  68133 

68142 

68151 

68160 

68169 

68178 

68187 

68196 

68  205 

481 

68  215  68  224 

68  233 

68  242 

68  251 

68  260 

68  269 

68  278 

68  2S7 

68  296 

482 

68  305  68  314 

68  323 

68  332 

68  341 

68  350 

68  359 

68  368 

68  377 

68  386 

483 

68  395  6S404 

68  413 

68  422 

68  431 

68  440 

68  449 

68  458 

68  467 

68  476 

484 

6S485  68  494 

68  502 

68  511 

68  520 

6S529 

68  538 

68  547 

68  556 

68  565 

485 

6S574  68  583 

68  592 

68  601 

68  610 

68  619 

68  628 

68  637 

68  646 

68  655 

486 

6S664  68  673' 

68  6S1 

68  690 

6S699 

68  708 

68  717 

68  726 

68  735 

68  744 

487 

68  753  68  762 

68  771 

6S780 

68  789 

68  797 

68  806 

6S815 

6S824 

68  833 

488 

6S842  68  851 

68S60 

6S869 

68  878 

6S8S6 

68  895 

6S904 

68  913 

68  922 

489 

68  931  68  940 

68  949 

68  958 

68  966 

68  975 

68  984 

68  993 

69  002 

69  011 

490 

69  020  69  02S 

69  037 

69  046 

69  055 

69  064 

69  073 

690S2 

69  090 

69099 

491 

69  108  69117 

69126 

69135 

69144 

69152 

69161 

69170 

69179 

691S8 

492 

69197  69  205 

69  214 

69  223 

69  232 

69  241 

69  249 

69  258 

69  267 

69  276 

493 

69  2S5  69  294 

69  302 

69  311 

69  320 

69  329 

69  338 

69  346 

69  355 

69  364 

494 

69  373  69  381 

69  390 

69  399 

69  408 

69  417 

69  425 

69  434 

69  443 

69  452 

495 

69  461  69  469 

69  478 

69  487 

69496 

69  504 

69  513 

69  522 

69  531 

69  539 

496 

69  548  69  557 

69  566 

69  574 

69  5S3 

69  592 

69  601 

69  609 

69  618 

69  627 

497 

69  636  69  644 

69  653 

69  662 

69  671 

69  679 

69  6S8 

69  697 

69  705 

69  714 

498 

69  723  69  732 

69  740 

69  749 

69  758 

69  767 

69  775 

69  784 

69  793 

69  801 

499 

69  810  69  819 

69  827 

69  836 

69  845 

69  854 

69  862 

69  871 

69  880 

69  888 

500 

69  897  69  906 

69  914 

69  923 

69932 

69  940 

69949 

69  958 

69  966 

69975 

N 

O    1 

2 

3 

4 

5 

6 

7 

8 

9 

450-500 


10 

500-550 

N 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

500 

69  897 

69  906 

69  914 

69  923 

69  932 

69  940 

69  949 

69  95S 

69  966 

69  975 

501 

69  984 

69  992 

70  001 

70  010 

70  018 

70027 

70  036 

70  044 

70  053 

70062 

502 

70  070 

70  079 

70  088 

70  096 

70105 

70114 

70122 

70131 

70140 

70148 

503 

70  157 

70165 

70  174 

70183 

70191 

70  200 

70  209 

70  217 

70  226 

70  234 

504 

70  243 

70  252 

70  260 

70  269 

70  278 

70  286 

70  295 

70  303 

70  312 

70  321 

505 

70  329 

70  338 

70  346 

70  355 

70  364 

70  372 

70  381 

70  389 

70  398 

70  406 

506 

70  415 

70  424 

70  432 

70  441 

70  449 

70  458 

70  467 

70  475 

70  484 

70  492 

507 

70  501 

70  509 

70  518 

70  526 

70  535 

70  544 

70  552 

70  561 

70  569 

70  578 

508 

70  586 

70  595 

70  603 

70  612 

70  621 

70  629 

70  638 

70  646 

70  655 

70  663 

509 

70  672 

70  680 

70  689 

70  697 

70  706 

70  714 

70  723 

70  731 

70  740 

70  749 

510 

70  757 

70  766 

70  774 

70  783 

70  791 

70  800 

70  808 

70  SI  7 

70  825 

70  834 

511 

70  842 

70  851 

70  859 

70  86S 

70S76 

70885 

70  893 

70  902 

70  910 

70  919 

512 

70  927 

70  935 

70  944 

70  952 

70  961 

70969 

70  978 

70  986 

70  995 

71  003 

513 

71012 

71020 

71029 

71037 

71046 

71054 

71063 

71071 

71079 

71  OSS 

514 

71096 

71105 

71113 

71  122 

71130 

71139 

71147 

71 155 

71164 

71172 

515 

71181 

71189 

71198 

71206 

71214 

71223 

71231 

71240 

71248 

71257 

516 

71265 

71273 

712S2 

71  290 

71299 

71307 

71315 

71324 

71332 

71341 

517 

71349 

71357 

71366 

71374 

71383 

71391 

71399 

7140S 

71416 

71425 

518 

71433 

71441 

71  450 

71  458 

71466 

71475 

714S3 

71492 

71  500 

71508 

519 

71517 

71525 

71533 

71542 

71550 

71559 

71567 

71575 

715S4 

71592 

520 

71600 

71609 

71617 

71625 

71634 

71642 

71650 

71659 

71667 

71675 

521 

71684 

71  692 

71700 

71709 

71717 

71725 

71734 

71742 

71750 

71759 

522 

71767 

71775 

717S4 

71792 

71S00 

71S09 

71817 

71S25 

71834 

71S42 

523 

71  850 

71  85S 

71867 

71875 

718S3 

71892 

71900 

71908 

71917 

71925 

524 

71933 

71941 

71950 

71958 

71966 

71975 

719S3 

71991 

71999 

72  008 

525 

72  016 

72  024 

72  032 

72  041 

72  049 

72  057 

72  066 

72  074 

72  082 

72  090 

526 

72  099 

72  107 

72  115 

72  123 

72  132 

72  140 

72  148 

72  156 

72  165. 

72173 

527 

72  181 

72  1S9 

72198 

72  206 

72  214 

72  222 

72  230 

72  239 

72  247 

72  255 

528 

72  263 

72  272 

72  2S0 

72  2S8 

72  296 

72  304 

72  313 

72  321 

72  329 

72  337 

529 

72  346 

72  354 

72  362 

72  370 

72  378 

72  3S7 

72  395 

72  403 

72  411 

72  419 

530 

72  428 

72  436 

72  444 

72  452 

72  460 

72  469 

72  477 

72  485 

72  493 

72  501 

531 

72  509 

72  518 

72  526 

72  534 

72  542 

72  550 

72  55S 

72  567 

72  575 

72  5S3 

532 

72  591 

72  599 

72  607 

72  616 

72  624 

72  632 

72  640 

72  648 

72  656 

72  665 

533 

72  673 

72  6S1 

72  6S9 

72  697 

72  705 

72  713 

72  722 

72  730 

72  738 

72  746 

534 

72  754 

72  762 

72  770 

72  779 

72  787 

72  795 

72  803 

72  811 

72  819 

72  827 

535 

72  835 

72  843 

72  852 

72  860 

72  868 

72  876 

72  884 

72  892 

72  900 

72  908 

536 

72  916 

72  925 

72  933 

72  941 

72  949 

72  957 

72  965 

72  973 

72  981 

72  989 

537 

72  997 

73  006 

73  014 

73  022 

73  030 

73  038 

73  046 

73  054 

73  062 

73  070 

538 

73  078 

73  0S6 

73  094 

73  102 

73  111 

73  119 

73  127 

73  135 

73  143 

73  151 

539 

73  159 

73  167 

73175 

73  183 

73  191 

73199 

73  207 

73  215 

73  223 

73  231 

540 

73  239 

73  247 

73  255 

73  263 

73  272 

73  280 

73  288 

73  296 

73  304 

73  312 

541 

73  320 

73  328 

73  336 

73  344 

73  352 

73  360 

73  368 

73  376 

73  3S4 

73  392 

542 

73  400 

73  408 

73  416 

73  424 

73  432 

73  440 

73  448 

73  456 

73  464 

73  472 

543 

73  4S0 

73  488 

73  496 

73  504 

73  512 

73  520 

73  528 

73  536 

73  544 

73  552 

544 

73  560 

73  563 

73  576 

73  5S4 

73  592 

73  600 

73  608 

73  616 

73  624 

73  632 

545 

73  640 

73  64S 

73  656 

73  664 

73  672 

73  679 

73  6S7 

73  695 

73  703 

73  711 

546 

73  719 

73  727 

73  735 

73  743 

73  751 

73  759 

73  767 

73  775 

73  7S3 

73  791 

547 

73  799 

73S07 

73S15 

73  823 

73S30 

73S38 

73  846 

73  854 

73  862 

73  870 

548 

73  878 

73SS6 

73  894 

73  902 

73  910 

73  918 

73  926 

73  933 

73  941 

73  949 

549 

73  957 

73  965 

73  973 

73  9S1 

73  9S9 

73  997 

74  005 

74  013 

74  020 

74  028 

550 

74  036 

74  044 
1 

74  052 

74  060 

74  06S 

74  076 

74  0S4 

74  092 

74  099 

74107 

X 

O 

.2 

3 

4 

5 

6 

7 

8 

9 

500-550 


550-600 


ii 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

550 

74  036 

74  044 

74  052 

74  060 

74  068 

74  076 

74  084 

74  092 

74  099 

74107 

551 

74115 

74123 

74131 

74139 

74147 

74  155 

74162 

74170 

74178 

74186 

552 

74194 

74  202 

74  210 

74  218 

74  225 

74  233 

74  241 

74  249 

74  257 

74  265 

553 

74  273 

74  280 

74  288 

74  296 

74  304 

74  312 

74  320 

74  327 

74  335 

74  343 

554 

74  351 

74  359 

74  367 

74  374 

74  382 

74  390 

74  398 

74  406 

74  414 

74  421 

555 

74429 

74  437 

74  445 

74  453 

74  461 

74  468 

74  476 

74  484 

74  492 

74  500 

556 

74  507 

74  515 

74  523 

74  531 

74  539 

74  547 

74  554 

74  562 

74  570 

74  578 

557 

74  586 

74  593 

74  601 

74  609 

74  617 

74  624 

74  632 

74  640 

74  648 

74  656 

558 

74  663 

74  671 

74  679 

74  687 

74  695 

74  702 

74  710 

74  718 

74  726 

74  733 

559 

74  741 

74  749 

74  757 

74  764 

74  772 

74  780 

74  788 

74  796 

74  803 

74  811 

560 

74  819 

74  827 

74  834 

74  842 

74  850 

74  858 

74  865 

74  873 

74  881 

74  889 

561 

74  896 

74  904 

74  912 

74  920 

74  927 

74  935 

74  943 

74  950 

74  958 

74  966 

562 

74  974 

74  981 

74  989 

71997 

75  005 

75  012 

75  020 

75  028 

75  035 

75  043 

563 

75  051 

75  059 

75  066 

75  074 

75  082 

75  089 

75  097 

75  105 

75  113 

75  120 

564 

75  128 

75  136 

75143 

75  151 

75159 

75  166 

75  174 

75  182 

75  189 

75  197 

565 

75  205 

75  213 

75  220 

75  228 

75  236 

75  243 

75  251 

75  259 

75  266 

75  274 

566 

75  282 

75  289 

75  297 

75  305 

75  312 

75  320 

75  328 

75  335 

75  343 

75  351 

567 

75  358 

75  366 

75  374 

75  381 

75  389 

75  397 

75  404 

75  412 

75  420 

75  427 

568 

75  435 

75  442 

75  450 

75  458 

75  465 

75  473 

75  481 

75  488 

75  496 

75  504 

569 

75  511 

75  519 

75  526 

75  534 

75  542 

75  549 

75  557 

75  565 

75  572 

75  580 

570 

75  587 

75  595 

75  603 

75  610 

75  618 

75  626 

75  633 

75  641 

75  648 

75  656 

571 

75  664 

75671 

75  679 

75  686 

75  694 

75  702 

75  709 

75  717 

75  724 

75  732 

572 

75  740 

75  747 

75  755 

75  762 

75  770 

75  778 

75  785 

75  793 

75  800 

75  808 

573 

75  815 

75  823 

75  831 

75  838 

75  846 

75  853 

75  861 

75  868 

75  876 

75  884 

574 

75  891 

75  899 

75  906 

75  914 

75  921 

75  929 

75  937 

75  944 

75  952 

75  959 

575 

75  967 

75  974 

75  982 

75  989 

75  997 

76005 

76012 

76020 

76027 

76035. 

576 

76  042 

76050 

76  057 

76  065 

76072 

76  080 

76  087 

76  095 

76103 

76110 

577 

76118 

76125 

76133 

76140 

76148 

76155 

76163 

76170 

76178 

76185 

578 

76193 

76  200 

76  208 

76  21* 

76  223 

76  230 

76  238 

76  245 

76  253 

76  260 

579 

76  268 

76  275 

76  283 

76  290 

76  298 

76  305 

76313 

76  320 

76  328 

76335 

580 

76  343 

76  350 

76  358 

76  365 

76  373 

76  380 

76  388 

76  395 

76403 

76410 

581 

76418 

76  425 

76  433 

76  440 

76  448 

76  455 

76  462 

76  470 

76  477 

76  485 

582 

76  492 

76  500 

76  507 

76  515 

76  522 

76  530 

76  537 

76  545 

76  552 

76  559 

583 

76  567 

76  574 

76  582 

76  589 

76  597 

76  604 

76  612 

76  619 

76  626 

76  634 

584 

76  641 

76  649 

76  656 

76664 

76671 

76  678 

76  686 

76  693 

76  701 

76  708 

585 

76  716 

76  723 

76  730 

76  738 

76  745 

76  753 

76  760 

76  768 

76  775 

76  782 

586 

76  790 

76  797 

76  805 

76  812 

76  819 

76  827 

76  834 

76  842 

76  849 

76  856 

587 

76  864 

76  871 

76  879 

76  886 

76  893 

76  901 

76908 

76  916 

76  923 

76  930 

588 

76  938 

76  945 

76  953 

76960 

76967 

76975 

76982 

76989 

76997 

77  004 

589 

77012 

77  019 

77  026 

77  034 

77041 

77048 

77  056 

77  063 

77  070 

77  078 

590 

77  085 

77  093 

77100 

77107 

77115 

77122 

77129 

77137 

77144 

77151 

591 

77159 

77166 

77173 

77181 

77188 

77195 

77  203 

77  210 

77  217 

77  225 

592 

77  232 

77  240 

77  247 

77  254 

77  262 

77  269 

77  276 

77  283 

77  291 

77  298 

593 

77  305 

77  313 

77  320 

77  327 

77  335 

77  342 

77  349 

77  357 

77  364 

77  371 

594 

77  379 

77  386 

77  393 

77  401 

77  408 

77  415 

77  422 

77  430 

77  437 

77  444 

595 

77  452 

77  459 

77  466 

77  474 

77  481 

77  488 

77  495 

77  503 

77  510 

77  517 

596 

77  525 

77  532 

77  539 

77  546 

77  554 

77  561 

77  568 

77  576 

77  583 

77  590 

597 

77  597 

77  605 

77  612 

77  619 

77  627 

77  634 

77  641 

77  648 

77  656 

77663 

598 

77  670 

77  677 

77  685 

77  692 

77  699 

77  706 

77  714 

77  721 

77  728 

77  735 

599 

77  743 

77  750 

77  757 

77  764 

77  772 

77  779 

77  786 

77  793 

77  801 

77  808 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77851 

77  859 

77  866 

77  873 

77  880 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

550-600 


12 

600-650 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851 

77  859 

77  866 

77  873 

77  880 

601 

77  887 

77  895 

77  902 

77  909 

77  916 

77  924 

77  931 

77  938 

77  945 

77  952 

602 

77  960 

77  967 

77  974 

77  981 

77  988 

77  996 

78  003 

78  010 

78  017 

78  025 

603 

78  032 

78  039 

78  046 

78  053 

78  061 

78  068 

78  075 

78  082 

78  089 

78  097 

604 

78104 

78111 

78118 

78125 

78132 

78140 

78147 

78154 

78161 

78168 

605 

78176 

78183 

78190 

78197 

78  204 

78  211 

78  219 

78  226 

78  233 

78  240 

606 

78  247 

78  254 

78  262 

78  269 

78  276 

78  283 

78  290 

78  297 

78  305 

78  312 

607 

78  319 

78  326 

78  333 

78  340 

78  347 

78  355 

78  362 

78  369 

78  376 

78  383 

608 

78  390 

78  398 

78  405 

78  412 

78  419 

78  426 

78  433 

78  440 

78  447 

78  455 

609 

78  462 

78  469 

78  476 

78  483 

78  490 

78  497 

78  504 

78  512 

78  519 

78  526 

610 

78  533 

78  540 

78  547 

78  554 

78  561 

78  569 

78  576 

78  583 

78  590 

78  597 

611 

78  604 

78  611 

78  618 

78  625 

78  633 

78  640 

78  647 

78  654 

78  661 

78  668 

612 

78  675 

78  682 

78  689 

78  696 

78  704 

78  711 

78  718 

78  725 

78  732 

78  739 

613 

78  746 

78  753 

78  760 

78  767 

78  774 

78  781 

78  789 

78  796 

78  803 

78  810 

614 

78  817 

78  824 

78  831 

78  838 

78  845 

78  852 

78  859 

78  866 

78  873 

78  880 

615 

78  888 

78  895 

78  902 

78  909 

78  916 

78  923 

78  930 

78  937 

78  944 

78  951 

616 

78  958 

78  965 

78  972 

78  979 

78  986 

78  993 

79  000 

79  007 

79  014 

79  021 

617 

79  029 

79  036 

79  043 

79  050 

79  057 

79  064 

79  071 

79  078 

79  085 

79  092 

618 

79  099 

79106 

79113 

79120 

79127 

79134 

79141 

79  148 

79155 

79162 

619 

79169 

79176 

791S3 

79190 

79197 

79  204 

79  211 

79  218 

79  225 

79  232 

620 

79  239 

79  246 

79  253 

79  260 

79  267 

79  274 

79  281 

79  288 

79  295 

79  302 

621 

79  309 

79316 

79  323 

79  330 

79  337 

79  344 

79  351 

79  358 

79  365 

79  372 

622 

79379 

79  386 

79  393 

79  400 

79  407 

79  414 

79  421 

79  428 

79  435 

79  442 

623 

79449 

79  456 

79463 

79  470 

79  477 

79  484 

79  491 

79  498 

79  505 

79  511 

624 

79  518 

79  525 

79  532 

79  539 

79  546 

79  553 

79  560 

79  567 

79  574 

79  581 

625 

79  588 

79  595 

79602 

79  609 

79  616 

79623 

79  630 

79  637 

79  644 

79  650 

626 

79  657 

79  664 

79  671 

79  678 

79  685 

79  692 

79  699 

79  706 

79  713 

79  720 

627 

79  727 

79  734 

79  741 

79  748 

79  754 

79  761 

79  768 

79  775 

79  782 

79  789 

628 

79  796 

79  803 

79  810 

79  817 

79  824 

79  831 

79  837 

79  844 

79  851 

79  858 

629 

79  865 

79  872 

79  879 

79  8S6 

79  893 

79  900 

79  906 

79  913 

79920 

79  927 

630 

79  934 

79941 

79948 

79955 

79962 

79969 

79975 

79  982 

79  989 

79  996 

631 

80  003 

80010 

80017 

80  024 

80  030 

80  037 

80  044 

80  051 

80  058 

80  065 

632 

80  072 

80  079 

80  085 

80  092 

80  099 

80106 

80113 

80120 

80127 

80134 

633 

80140 

80147 

80154 

80161 

80168 

80175 

80182 

80188 

80195 

80  202 

634 

80  209 

80  216 

80  223 

80  229 

80  236 

80  243 

80  250 

80  257 

80  264 

80  271 

635 

80  277 

80  284 

80  291 

80  298 

80  305 

80  312 

80  318 

80  325 

80332 

80  339 

636 

80  346 

80  353 

80  359 

80  366 

80  373 

80  380 

80  387 

80  393 

80  400 

80  407 

637 

80  414 

80  421 

80  428 

80  434 

80  441 

80  448 

80  455 

80  462 

80  468 

80  475 

638 

80  482 

80  489 

80496 

80  502 

80  509 

80  516 

80  523 

80  530 

80  536 

80  543 

639 

80  550 

80  557 

80  564 

80  570 

80  577 

80  584 

80  591 

80  598 

80  604 

80  611 

640 

80  618 

80  625 

80  632 

80  638 

80  645 

80  652 

80  659 

80  665 

80  672 

80  679 

641 

80  686 

80  693 

80  699 

80  706 

80  713 

80  720 

80  726 

80  733 

80  740 

80  747 

642 

80  754 

80  760 

80  767 

80  774 

80  781 

80  787 

80  794 

80  801 

80  808 

80  814 

643 

80  821 

80  828 

80  835 

80  841 

80  848 

80  855 

80  862 

80  868 

80  875 

80  882 

644 

80889 

80  895 

80  902 

80909 

80  916 

80922 

80  929 

80  936 

80  943 

80  949 

645 

80956 

80963 

80  969 

80976 

80  983 

80990 

80  996 

81003 

81010 

81017 

646 

81023 

81030 

81037 

81043 

81050 

81057 

81064 

81070 

81077 

81084 

647 

81090 

81097 

81104 

81111 

81117 

81124 

81131 

81137 

81144 

81151 

648 

81158 

81164 

81171 

81178 

81184 

81191 

81198 

81204 

81211 

81218 

649 

81224 

81231 

81238 

81245 

81251 

81258  81265 

81271 

81278 

81285 

650 

81291 

81298  81305 

81311 

81318 

81325 

81331 

81338 

81345 

81351 

K 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600-650 


650-7 

'00 

13 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

650 

81291 

81298  81305 

81311 

81318 

81325 

81331 

81338 

81345 

81351 

651 

81358 

81365 

81371 

81378 

81385 

81391 

81398 

81405 

81411 

81418 

652 

81425 

81431 

81438 

81445 

81451 

81458  81465 

81471 

81478 

81485 

653 

81491 

81498 

81505 

81511 

81518 

81525 

81531 

81538 

81544 

81551 

654 

81558 

81564 

81571 

81578 

81584 

81591 

81598 

81604 

81611 

81617 

655 

81624 

81631 

81637 

81644 

81651 

81657 

81664 

81671 

81677 

81684 

656 

81690 

81697 

81704 

81710 

81717 

81723 

81730 

81737 

81743 

81750 

657 

81757 

81763 

81770 

81776 

81783 

81790 

81796 

81803 

81809 

81816 

658 

81823 

81829 

81836 

81842 

81849 

81856 

81862 

81869 

81875 

81882 

659 

81889 

81895 

81902 

81908 

81915 

81  921 

81928 

81935 

81941 

81948 

660 

81954 

81961 

81968 

81974 

81981 

81987 

81994 

82  000 

82  007 

82  014 

661 

82  020 

82  027 

82  033 

82  040 

82  046 

82  053 

82  060 

82  066 

82  073 

82  079 

662 

82  086 

82  092 

82  099 

82  105 

82112 

82119 

82125 

82132 

82138 

82145 

663 

82151 

82  158 

82164 

82  171 

82  178 

82184 

82191 

82197 

82  204 

82  210 

664 

82  217 

82  223 

82  230 

82  236 

82  243 

82  249 

82  256 

82  263 

82  269 

82  276 

665 

82  282 

82  289 

82  295 

82  302 

82  308 

82  315 

82  321 

82  328 

82  334 

82  341 

666 

82  347 

82  354 

82  360 

82  367 

82  373 

82  380 

82  387 

82  393 

82  400 

82  406 

667 

82  413 

82  419 

82  426 

82  432 

82  439 

82  445 

82  452 

82  458 

82  465 

82  471 

668 

82  478 

82  484 

82  491 

82  497 

82  504 

82  510 

82  517 

82  523 

82  530 

82  536 

669 

82  543 

82  549 

82  556 

82  562 

82  569 

82  575 

82  582 

82  588 

82  595 

82  601 

670 

82  607 

82  614 

82  620 

82  627 

82  633 

82  640 

82  646 

82  653 

82  659 

82  666 

671 

82  672 

82  679 

82  685 

82  692 

82  698 

82  705 

82  711 

82  718 

82  724 

82  730 

672 

82  737 

82  743 

82  750 

82  756 

82  763 

82  769 

82  776 

82  782 

82  789 

82  795 

673 

82  802 

82  808 

82  814' 

82  821 

82  827 

82  834 

82  840 

82  847 

82  853 

82  860 

674 

82  866 

82  872 

82  879 

82  885 

82  892 

82  898 

82  905 

82  911 

82  918 

82  924 

675 

82  930 

82  937 

82  943 

82  950 

82  956 

82  963 

82  969 

82975 

82  982 

82  988 

676 

82  995 

83  001 

83  008 

83  014 

83  020 

83  027 

83  033 

83  040 

83  046 

83  052 

677 

83  059 

83  065 

83  072 

83  078 

83  085 

83  091 

83  097 

83104 

83110 

83117 

678 

83  123 

83  129 

83  136 

83  142 

83  149 

83155 

83161 

83168 

83  174 

83181 

679 

83187 

•83193 

83  200 

83  206 

83  213 

83  219 

83  225 

83  232 

83  238  83  245 

680 

83  251 

83  257 

83  264 

83  270 

83  276 

83  283 

83  289 

83  296 

83  302 

83  308 

681 

83  315 

83  321 

83  327 

83  334 

83  340 

83  347 

83  353 

83  359 

83  366 

83  372 

682 

83  378 

83  385 

83  391 

83  398 

83  404 

83  410 

83  417 

83  423 

83  429 

83  436 

683 

83  442 

83  448 

83  455 

83  461 

83  467 

83  474 

83  480 

83  487 

83  493 

83  499 

684 

83  506 

83  512 

83  518 

83  525 

83  531 

83  537 

83  544 

83  550 

83  556 

83  563 

685 

83  569 

83  575 

83  582 

83  588 

83  594 

83  601 

83  607 

83  613 

83  620 

83  626 

686 

83  632 

83  639 

83  645 

83  651 

83  658 

83  664 

83  670 

83  677 

83  683 

83  689 

687 

83  696 

83  702 

83  708 

83  715 

83  721 

83  727 

83  734 

83  740 

83  746 

83  753 

688 

83  759 

83  765 

83  771 

83  778 

83  784 

83  790 

83  797 

83  803 

83  809 

83  816 

689 

83  822 

83  828 

83  835 

83  841 

83  847 

83  853 

83  860 

83  866 

83  872 

83  879 

690 

83  885 

83  891 

83  897 

83  904 

83  910 

83  916 

83  923 

83  929 

83  935 

83  942 

691 

83  948 

83  954 

83  960 

83  967 

83  973 

83  979 

83  985 

83  992 

83  998 

84  004 

692 

84  011 

84  017 

84  023 

84  029 

84  036 

84  042 

84  048 

84055 

84  061 

84  067 

693 

84  073 

84  080 

84  086 

84  092 

84  098 

84105 

84111 

84117 

84123 

84130 

694 

84136 

84142 

84148 

84155 

84161 

84167 

84173 

84180 

84186 

84192 

695 

84198 

84  205 

84  211 

84  217 

84  223 

84  230 

84  236 

84  242 

84  248 

84  255 

696 

84  261 

84  267 

84  273 

84  280 

84  286 

84  292 

84  298 

84  305 

84  311 

84  317 

697 

84  323 

84  330 

84  336 

84  342 

84  348 

84  354 

84  361 

84  367 

84  373 

84379 

698 

84  386 

84  392 

84  398 

84  404 

84  410 

84  417 

84  423 

84  429 

84  435 

84  442 

699 

84  448 

84  454 

84  460 

84  466 

84  473 

84  479 

84  485 

84  491 

84  497 

84  504 

700 

S4  510 

84  516 

84  522 

84  528 

84  535 

84  541 

84  547 

84  553 

84  559 

84  566 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

650-700 


14 

700-750 

N 

O 

1 

2 

3 

4 

5 

6 

7 

84  553 

8 

9 

700 

84  510 

84  516  84  522  84  528  84.535 

84  541 

84  547 

84  559 

84  566 

701 

84  572 

84  578 

84  584 

84  590 

84  597 

84  603 

84  609 

84  615 

84  621 

84  628 

702 

84  634 

84  640 

84  646 

84  652 

84  658 

84  665 

84  671 

84  677 

84  683 

84  689 

703 

84  696 

84  702 

84  708 

84  714 

84  720 

84  726 

84  733 

84  739 

84  745 

84  751 

704 

84  757 

84  763 

84  770 

84  776 

84  782 

84  788 

84  794 

84  800 

84  807 

84  813 

705 

84  819 

84  825 

84  831 

84  837 

84  844 

84  850 

84  856 

84  862 

84  868 

84  874 

706 

84  880 

84  887 

84  893 

84  899 

84  905 

84  911 

84  917 

84  924 

84  930 

84  936 

707 

84  942 

84  948 

84  954 

84  960 

84  967 

84  973 

84  979 

84  985 

84  991 

84  997 

708 

85  003 

85  009 

85  016 

85  022 

85  028 

85  034 

85  040 

85  046 

85  052 

85  058 

709 

85  065 

85  071 

85  077 

85  083 

85  089 

85  095 

85  101 

85  107 

85  114 

85  120 

710 

85  126 

85132 

85  138 

85144 

85150 

85  156 

85  163 

85  169 

85175 

85  181 

711 

85  187 

85  193 

85  199 

85  205 

85  211 

85  217  "85  224 

85  230 

85  236 

85  242 

712 

85  248 

85  254 

85  260 

85  266 

85  272 

85  278 

85  285 

85  291 

85  297 

85  303 

713 

85  309 

85  315 

85  321 

85  327 

85  333 

85  339 

85  345 

85  352 

85  358 

85  364 

714 

85  370 

85  376 

85  382 

85  388 

85  394 

85  400 

85  406 

85  412 

85  418 

85  425 

715 

85  431 

85  437 

85  443 

85  449  85  455 

85  461 

85  467 

85  473 

85  479 

85  485 

716 

85  491 

85  497 

85  503 

85  509 

85  516 

85  522 

85  528 

85  534 

85  540 

85  546 

717 

85  552 

85  558 

85  564 

85  570 

85  576 

85  582 

85  588 

85  594 

85  600 

85  606 

718 

85  612 

85  618 

85  625 

85  631 

85  637 

85  643 

85  649 

85  655 

85  661 

85  667 

719 

85  673 

85  679 

85  685 

85  691 

85  697 

85  703 

85  709 

85  715 

85  721 

85  727 

720 

85  733 

85  739 

85  745 

85  751 

85  757 

85  763 

85  769 

85  775 

85  781 

85  788 

721 

85  794 

85  800 

85  806 

85  812 

85  818 

85  824 

85  830 

85  836 

85  842 

85  848 

722 

85  854 

85  860 

85  866 

85  872 

85  878 

85  884 

85  890 

85  896 

85  902 

85  908 

723 

85  914 

85  920 

85  926 

85  932 

85  938 

85  944 

85  950 

85  956 

85  962 

85  968 

724 

85  974 

85  980 

85  986 

85  992 

85  998 

86004 

86010 

86  016 

86022 

86028 

725 

86034 

86040 

86046 

86052 

86058 

86  064 

86070 

86076 

86  082 

86088 

726 

86  094 

86100 

86106 

86112 

86118 

86124 

86130 

86136 

86141 

86147 

727 

86153 

86159 

86165 

86171 

86177 

86183 

86189 

86195 

86  201 

86  207 

728 

86  213 

86  219 

86  225 

86  231 

86  237 

86  243 

86  249 

86  255 

86  261 

86  267 

729 

86  273 

86  279  86  285 

86291 

86  297 

86  303 

86  308 

86314' 

86  320 

86  326 

730 

86  332 

86  338 

86  344 

86  350 

86  356 

86  362 

86  368 

86374 

86  380 

86386 

731 

86  392 

86  398 

86  404 

86410 

86  415 

86  421 

86  427 

86433 

86  439 

86  445 

732 

86  451 

86  457 

86  463 

86  469  86  475 

86  4S1 

86  487 

86  493 

86  499 

86  504 

733 

86  510 

86  516 

86  522 

86  528 

86  534 

86  540 

86  546 

86  552 

86  558 

86  564 

734 

86  570 

86  576 

86  581 

86  587 

86  593 

86  599 

86  605 

86611 

86  617 

86  623 

735 

86  629 

86  635 

86641 

86  646 

86  652 

86  658 

86  664 

86670 

86  676 

86  682 

736 

86688 

86  694 

86  700 

86  705 

86  711 

86  717 

86  723 

86  729 

86  735 

86  741 

737 

86  747 

86  753 

86  759 

86  764 

86  770 

86  776 

86  782 

86  788 

86  794 

86  800 

738 

86  806 

86  812 

86  817 

86  823 

86  829 

86  835 

86  841 

86  847 

86  853 

86  859 

739 

86  864 

86  870 

86  876 

86  882 

86  888 

86  894 

86  900 

86906 

86  911 

86917 

740 

86923 

86929  86  935 

86  941 

86  947 

86953 

86  958 

86964 

86  970 

86976 

741 

86982 

86988 

86  994 

86  999 

87  005 

87  011 

87  017 

87  023 

87  029 

87  035 

742 

87040 

87  046 

87  052 

87  058 

87  064 

87  070 

87  075 

87  081 

87  087 

87  093 

743 

87  099 

87105 

87111 

87116 

87122 

87128 

87134 

87140 

87146 

87151 

744 

87157 

87163 

87169  87175 

87181 

87186 

87192 

87198 

87  204 

87  210 

745 

87  216  87  221 

87  227 

87  233 

87  239 

87  245 

87  251 

87  256 

87  262 

87  268 

746 

87  274 

87  280 

87  286 

87  291 

87  297 

87  303 

87  309 

87  315 

87  320 

87  326 

747 

87  332 

87  338 

87  344 

87  349 

87  355 

87  361 

87  367 

87  373 

87  379 

87  384 

748 

87  390 

87  396 

87  402 

87  408 

87  413 

87  419 

87  425 

87  431 

87  437 

87442 

749 

87  448 

87  454 

87  460 

87  466 

87  471 

87  477 

87  483 

87  489 

87  495 

87  500 

750 

87  506 

87  512 

87  518 

87  523 

87  529 

87  535 

87  541 

87  547 

87  552 

87  558 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

700-750 


' 

750-800 

15 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

750 

87  506 

87  512 

87  518 

87  523 

87  529 

87  535 

87  541 

87  547 

87  552 

87  558 

751 

87  564 

87  570 

87  576 

87  581 

87  587 

87  593 

87  599 

87  604 

87  610 

87  616 

752 

87  622 

87  628 

87  633 

87  639 

87  645 

87  651 

87  656 

87  662 

87  668 

87  674 

753 

87  679 

87  6S5 

87  691 

87  697 

87  703 

87  708 

87  714 

87  720 

87  726 

87  731 

754 

87  737 

87  743 

87  749 

87  754 

87  760 

87  766 

87  772 

87  777 

87  783 

87  789 

755 

87  795 

87  800 

87  806 

87  812 

87  818 

87S23 

87  829 

87  835 

87  841 

87  846 

756 

87  852 

87  858 

87  864 

87  869 

87  875 

87  881 

87  887 

87  892 

87  898 

87  904 

757 

87  910 

87  915 

87  921 

87  927 

87  933 

87  938 

87  944 

87  950 

87  955 

87  961 

758 

87  967 

87  973 

87  978 

87  9S4 

87  990 

87  996 

88  001 

88  007 

88  013 

88  018 

759 

88  024 

88030 

88  036 

88  041 

88  047 

88  053 

88  058 

88  064 

88  070 

88  076 

760 

88  081 

88  087 

88  093 

88  098 

88104 

88110 

88116 

88121 

88127 

88133 

761 

88138 

88144 

88150 

88156 

88161 

88167 

88173 

88178 

88  184 

88190 

762 

88195 

88  201 

88  207 

88  213 

88  218 

88  224 

88  230 

88  235 

88  241 

88  247 

763 

88  252 

88  258 

88  264 

88  270 

88  275 

88  281 

88  287 

88  292 

88  298 

88  304 

764 

88  309 

88  315 

88  321 

88  326 

88  332 

88  338 

88  343 

88  349 

88  355 

88  360 

765 

88  366 

88  372 

88  377 

88  383 

88  389 

88  395 

88  400 

88  406 

88  412 

88  417 

766 

88  423 

88  429 

88  434 

88  440 

88  446 

88  451 

88  457 

88  463 

88  468 

88  474 

767 

88  4S0 

88  485 

88  491 

88  497 

88  502 

88  508 

88  513 

88  519 

88  525 

88  530 

76S 

88  536 

88  542 

88  547 

88  553 

88  559 

88  564 

88  570 

88  576 

88  581 

88  587 

769 

88  593 

88  598 

88  604 

88  610 

8S615 

88621 

88  627 

88  632 

88  638 

88  643 

770 

88  649 

88  655 

88  660 

88666 

88  672 

88  677 

88  683 

88  689 

88  694 

88  700 

771 

88  705 

88  711 

88  717 

88  722 

88  728 

88  734 

88  739 

88  745 

88  750 

88  756 

772 

88  762 

88  767 

88  773 

88  779 

88  784 

88  790 

88  795 

88  801 

88  807 

88  812 

773 

88  818 

88  824 

88  829 

88  835 

88  840 

88  846 

88  852 

88  857 

88  863 

88  868 

774 

88874 

88  880 

88  885 

88  891 

88  897 

88  902 

88908 

88  913 

88  919  88  925 

775 

88  930 

88  936 

88  941 

88  947 

88  953 

88  958 

88964 

88  969 

88  975 

88  981 

776 

88  986 

88  992 

88  997 

89  003 

89  009 

89  014 

89020 

89  025 

89  031 

89  037 

777 

89  042 

89  048 

89  053 

89059 

89064 

89  070 

89  076 

89  0S1 

89  087 

89  092 

778 

89  098 

89104 

89109 

89115 

89120 

89126 

89131 

89137 

89143 

89148 

779 

89154 

89159 

89165 

89170 

89176 

89182 

89187 

89193 

89198 

89  204 

780 

89  209 

89  215 

89  221 

89  226 

89  232 

89  237 

89  243 

89  248 

89  254 

89  260 

781 

89  265 

89  271 

89  276 

89  282 

89  287 

89  293 

89  298 

89304 

89  310 

89  315 

782 

89  321 

89  326 

89  332 

89  337 

89  343 

89  348 

89  354 

89  360 

89  365 

89  371 

783 

89  376 

89  382 

89  387 

89  393 

89  398 

89  404 

89  409  89415 

89421 

89  426 

784 

89432 

89  437 

89  443 

89  448 

89454 

89  459 

89  465 

89  470 

89476 

89481 

785 

89  487 

89492 

89  498 

89  504 

89  509 

89  515 

89  520 

89  526 

89  531 

89  537 

786 

89  542 

89  548 

89  553 

89  559 

89  564 

89  570 

89  575 

89  581 

89  586 

89  592 

787 

89  597 

89  603 

89  609 

89  614 

89  620 

89  625 

89  631 

89636 

89  642 

89  647 

788 

89  653 

89  658 

89  664 

89  669 

89  675 

89  680 

89  686 

89  691 

89  697 

89  702 

789 

89  708 

89  713 

89  719 

89  724 

89  730 

89  735 

89  741 

89  746 

89  752 

89  757 

790 

89  763 

89  768 

89  774 

89  779 

89  785 

89  790 

89  796 

89  801 

89  807 

89  812 

791 

89  818 

89  823 

89  829 

89  834 

89  840 

89  845 

89  851 

89  856 

89  862 

89  867 

792 

89  873 

89  878 

89  883 

89  889 

89  894 

89  900 

89  905 

89  911 

89  916 

89  922 

793 

89  927 

89  933 

89  938 

89  944 

89  949 

89  955 

89  960 

89  966 

89  971 

89977 

794 

89  982 

89988 

89  993 

89  998 

90  004 

90009 

90  015 

90  020 

90  026 

90  031 

795 

90037 

90  042 

90  048 

90  053 

90  059 

90064  90  069  90  075 

90080 

90  086 

796 

90  091 

90  097 

90102 

90108 

90113 

90119 

90124 

90129 

90135 

90140 

797 

90146 

90151 

90157 

90  162 

90168 

90173 

90179 

90184 

90189 

90195 

798 

90  200 

90  206 

90  211 

90  217 

90  222 

90  227 

90  233 

90  238 

90  244 

90  249 

799 

90  255 

90  260 

90  266 

90  271 

90  276 

90  282 

90  287 

90  293 

90  298 

90  304 

800 

90  309 

90  314 

90  320 

90325 

90  331 

90  336 
5 

90  342 

90  347 

90  352 

90358 

N 

O 

1 

2 

3 

4 

6 

7 

8 

9 

750-800 


16 

800-850 

N 

O 

1 

2 

3 

4 

5 

6 

7    8 

9 

800 

90  309 

90  314 

90  320 

90  325 

90  331 

90  336 

90  342 

90  347  90  352 

90  358 

801 

90  363 

90  369 

90  374 

90  380 

90  385 

90  390 

90  396 

90  401  90  407 

90  412 

802 

90  417 

90  423 

90  428 

90  434 

90  439 

90  445 

90  450 

90  455  90  461 

90  466 

803 

90  472 

90  477 

90  482 

90  488 

90  493 

90  499 

90  504 

90  509  90  515 

90  520 

804 

90  526 

90  531 

90  536 

90  542 

90  547 

90  553 

90  558 

90  563  90  569 

90  574 

805 

90  580  90  585 

90  590 

90  596 

90  601 

90  607 

90  612 

90  617  90  623 

90  628 

806 

90  634 

90  639 

90  644 

90  650 

90  655 

90  660 

90  666 

90  671  90  677 

90  682 

807 

90  687 

90  693 

90  698 

90  703 

90  709 

90  714 

90  720 

90  725  90  730 

90  736 

808 

90  741 

90  747 

90  752 

90  757 

90  763 

90768 

90  773 

90  779  90  784 

90  789 

809 

90  795 

90  800 

90  806 

90  811 

90  816 

90  822 

90  827 

90  832  90  838 

90  843 

810 

90  849 

90  854 

90  859 

90  865 

90  870 

90  875 

90  881 

90  886  90  891 

90  897 

811 

90  902 

90  907 

90  913 

90  918 

90  924 

90  929 

90  934 

90  940  90  945 

90  950 

812 

90  956 

90  961 

90  966 

90  972 

90  977 

90  982 

90  988 

90  993  90  998 

91004 

813 

91009 

91014 

91020 

91025 

91030 

91036 

91041 

91046  91052 

91057 

814 

91062 

91068 

91073 

91078 

91084 

91089 

91094 

91  100  91  105 

91110 

815 

91116 

91121 

91126 

91132 

91137 

91142 

91148 

91153  91158 

91164 

816 

91169 

91174 

91180 

91185 

91190 

91196 

91201 

91206  91212 

91217 

817 

91222 

91228 

91233 

9123S 

91243 

91249 

91254 

91259  91265 

91270 

818 

91275 

91281 

91286 

91291 

91297 

91302 

91307 

91312  91318 

91323 

819 

91328 

91334 

91339 

91344 

91350 

91355 

91360 

91365  91371 

91376 

820 

91381 

91387 

.91392 

91397 

91403 

91408 

91413 

91418  91424 

91429 

821 

91434 

91440 

91445 

91450 

91455 

91461 

91466 

91471  91477 

91482 

822 

91487 

91492 

91498 

91503 

91508 

91514 

91519 

91524  91529 

91535 

823 

91540 

91545 

91  551 

91556 

91561 

91566 

91572 

91577  91582 

91587 

824 

91593 

91598 

91603 

91609 

91614 

91619  91624  91630  91635 

91640 

825 

91645 

91651 

91656 

91661 

91666 

91672 

91677 

91682  91687 

91693 

826 

91698 

91703 

91709 

91714 

91719 

91724 

91  730  91  735  91  740 

91745 

827 

91751 

91756 

91761 

91766 

91772 

91777 

91782 

91  787  91  793 

91798 

828 

91803 

91  808 

91814 

91819 

91824 

91829 

91834 

91840  91845 

91850 

829 

91855 

91861 

91866 

91871 

91876 

91882 

91887 

91892  91897 

91903 

830 

91908 

91913 

91918 

91924 

91929 

91934  91939  91944  91950  91955 

831 

91960 

91965 

91971 

91976 

91981 

91986 

91991 

91997  92  002 

92  007 

832 

92  012 

92  018 

92  023 

92  028 

92  033 

92  038 

92  044 

92  049  92  054 

92  059 

833 

92  065 

92  070 

92  075 

92  080 

92  085 

92  091 

92  096 

92  101  92  106 

92111 

834 

92117 

92122 

92127 

92132 

92137 

92  143 

92  148 

92  153  92158 

92163 

835 

92169 

92174 

92  179 

92184 

92  189 

92  195 

92  200 

92  205  92  210 

92  215 

836 

92  221 

92  226 

92  231 

92  236 

92  241 

92  247 

92  252 

92  257  92  262 

92  267 

837 

92  273 

92  278 

92  283 

92  288 

92  293 

92  298 

92  304 

92  309  92  314 

92  319 

838 

92  324 

92  330 

92  335 

92  340 

92  345 

92  350 

92  355 

92  361  92  366 

92  371 

839 

92  376 

92  381 

92  387 

92  392 

92  397 

92  402 

92  407 

92  412  92  418 

92  423 

840 

92  428 

92  433 

92  438 

92  443 

92  449 

92  454 

92  459 

92  464  92  469 

92  474 

841 

92  480 

92  485 

92  490 

92  495 

92  500 

92  505 

92  511 

92  516  92  521 

92  526 

842 

92  531 

92  536 

92  542 

92  547 

92  552 

92  557 

92  562 

92  567  92  572 

92  578 

843 

92  583 

92  588 

92  593 

92  598 

92  603 

92  609 

92  614 

92  619  92  624 

92  629 

844 

92  634 

92  639 

92  645 

92  650  92  655 

92  660 

92  665 

92  670  92  675 

92  681 

845 

92  686 

92  691 

92  696 

92  701 

92  706 

92  711 

92  716 

92  722  92  727 

92  732 

846 

92  737 

92  742 

92  747 

92  752 

92  758 

92  763 

92  768 

92  773  92  778 

92  783 

847 

92  788 

92  793 

92  799 

92  804 

92  809 

92  814 

92  819 

92  824  92  829 

92  834 

848 

92  840 

92  845 

92  850 

92  855 

92  860 

92  865 

92  870 

92  875  92  8S1 

92  886 

849 

92  891 

92  896 

92  901 

92  906 

92  911 

92  916 

92  921 

92  927  92  932 

92  937 

850 

92  942 

92  947 

92  952 

92  957 

92  962 

92  967 

92  973 

92  978  92  983 

92  988 

N 

O 

1 

2 

3 

4 

5 

6 

7    8 

9 

800-850 


850-900 


IT 


N 

O 

92  942 

1 

2 

3 

4 

5 

6 

7 

8 

9 

850 

92  947 

92  952 

92  957 

92  962 

92  967 

92  973 

92  978 

92  983 

92  988 

851 

92  993 

92  998 

93  003 

93  008 

93  013 

93  018 

93  024 

93  029 

93  034 

93  039 

852 

93  044 

93  049 

93  054 

93  059 

93  064 

93  069 

93  075 

93  080 

93  085 

93  090 

853 

93  095 

93  100 

93  105 

93110 

93  115 

93  120 

93  125 

93  131 

93  136 

93141 

854 

93146 

93  151 

93  156 

93161 

93  166 

93171 

93176 

93  181 

93  186 

93192 

855 

93197 

93  202 

93  207 

93  212 

93  217 

93  222 

93  227 

93  232 

93  237 

93  242 

856 

93  247 

93  252 

93  258 

93  263 

93  268 

93  273 

93  278 

93  283 

93  288 

93  293 

857 

93  298 

93  303 

93  308 

93  313 

93  318 

93  323 

93  328 

93  334 

93  339 

93  344 

858 

93  349 

93  354 

93  359 

93  364 

93  369 

93  374 

93  379 

93  384 

93  389 

93  394 

859 

93  399 

93  404 

93  409 

93  414 

93  420 

93  425 

93  430  93  435 

93  440 

93  445 

860 

93  450 

93  455 

93  460 

93  465 

93  470 

93  475 

93  480 

93  485 

93  490 

93  495 

861 

93  500 

93  505 

93  510 

93  515 

93  520 

93  526 

93  531 

93  536 

93  541 

93  546 

862 

93  551 

93  556 

93  561 

93  566 

93  571 

93  576 

93  581 

93  586 

93  591 

93  596 

863 

93  601 

93  606 

93  611 

93  616 

93  621 

93  626 

93  631 

93  636 

93  641 

93  646 

864 

93  651 

93  656 

93  661 

93  666 

93  671 

93  676 

93  682 

93  687 

93  692 

93  697 

865 

93  702 

93  707 

93  712 

93  717 

93  722 

93  727 

93  732 

93  737 

93  742 

93  747 

866 

93  752 

93  757 

93  762 

93  767 

93  772 

93  777 

93  782 

93  787 

93  792 

93  797 

867 

93  802 

93  807 

93  812 

93  817 

93  822 

93  827 

93  832 

93  837 

93  842 

93  847 

868 

93  852 

93  857 

93  862 

93  867 

93  872 

93  877 

93  882 

93  887 

93  892 

93  897 

869 

93  902 

93  907 

93  912 

93  917 

93  922 

93  927 

93  932 

93  937 

93  942 

93  947 

870 

93  952 

93  957 

93  962 

93  967 

93  972 

93  977 

93  982 

93  987 

93  992 

93  997 

871 

94  002 

94  007 

94  012 

94  017 

94  022 

94  027 

94  032 

94  037 

94  042 

94  047 

872 

94  052 

94  057 

94  062 

94  067 

94  072 

94  077 

94  082 

*  94  086 

94  091 

94  096 

873 

94101 

94106 

94111 

94  116 

94121 

94126 

94131 

94136 

94  141 

94146 

874 

94151 

94156 

94  161 

94166 

94171 

94176 

94181 

94186 

94191 

94196 

875 

94  201 

94  206 

94  211 

94  216 

94  221 

94  226 

94  231 

94  236 

94  240 

94  245 

876 

94  250 

94  255 

94  260 

94  265 

94  270 

94  275 

94  280 

94  285 

94  290 

94  295 

877 

94  300 

94  305 

94  310  94  315 

94  320 

94  325 

94  330 

94  335 

94  340 

94  345 

878 

94  349 

94  354 

94  359 

94  364 

94  369 

94  374 

94  379 

94  384 

94  389 

94  394 

879 

94  399 

94  404 

94  409 

94  414 

94  419 

94  424 

94  429 

94  433 

94  438 

94  443 

880 

94  448 

94  453 

94  458 

94  463 

94  468 

94  473 

94  478 

94  483 

94  488 

94  493 

881 

94  498 

94  503 

94  507 

94  512 

94  517 

94  522 

94  527 

94  532 

94  537 

94  542 

882 

94  547 

94  552 

94  557 

94  562 

94  567 

94  571 

94  576 

94  581 

94  586 

94  591 

883 

94  596 

94  601 

94  606 

94  611 

94  616 

94  621 

94  626 

94  630 

94  635 

94  640 

884 

94  645 

94  650 

94  655 

94  660 

94  665 

94  670 

94  675 

94  680 

94  685 

94  689 

885 

94  694 

94  699 

94  704 

94  709 

94  714 

94  719 

94  724 

94  729 

94  734 

94  738 

886 

94  743 

94  748 

94  753 

94  758 

94  763 

94  768 

94  773 

94  778 

94  783 

94  787 

887 

94  792 

94  797 

94  802 

94  807 

94  812 

94  817 

94  822 

94  827 

94  832 

94  836 

888 

94  841 

94  846 

94  851 

94  856 

94  861 

94  866 

94  871 

94  876 

94  880 

94  885 

889 

94  890 

94  895 

94  900 

94  905 

94  910 

94  915 

94  919 

94  924 

94  929 

94  934 

890 

94  939 

94  944 

94  949 

94  954 

94  959 

94  963 

94  968 

94  973 

94  978 

94  983 

891 

94  988 

94  993 

94  998 

95  002 

95  007 

95  012 

95  017 

95  022 

95  027 

95  032 

892 

95  036 

95  041 

95  046 

95  051 

95  056 

95  061 

95  066 

95  071 

95  075 

95  080 

893 

95  085 

95  090 

95  095 

95  100 

95  105 

95  109 

95  114 

95  119 

95  124 

95  129 

894 

95  134 

95  139 

95143 

95  148 

95  153 

95  158 

95  163 

95  168 

95  173 

95  177 

895 

95  182 

95  187 

95  192 

95  197 

95  202 

95  207 

95  211 

95  216 

95  221 

95  226 

896 

95  231 

95  236 

95  240 

95  245 

95  250 

95  255 

95  260 

95  265 

95  270 

95  274 

897 

95  279 

95  284 

95  289 

95  294 

95  299 

95  303 

95  308 

95  313 

95  318 

95  323 

898 

95  328 

95  332 

95  337 

95  342 

95  347 

95  352 

95  357 

95  361 

95  366 

95  371 

899 

95  376 

95  381 

95  386 

95  390 

95  395 

95  400 

95  405 

95  410 

95  415 

95  419 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

K 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

850-900 


18 

90 

10-950 

N  . 

O 

1    2 

3 

4 

5 

6 

7 

8 

9 

900 

95  424 

95  429  95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

901 

95  472 

95  477  95  482 

95  487 

95  492 

95  497 

95  501 

95  506 

95  511 

95  516 

902 

95  521 

95  525  95  530 

95  535 

95  540 

95  545 

95  550 

95  554 

95  559 

95  564 

903 

95  569 

95  574  95  578 

95  583 

95  588 

95  593 

95  598 

95  602 

95  607 

95  612 

904 

95  617 

95  622  95  626 

95  631 

95  636 

95  641 

95  646 

95  650 

95  655 

95  660 

905 

95  665 

95  670  95  674 

95  679 

95  684 

95  689 

95  694 

95  698 

95  703 

95  708 

906 

95  713 

95  718  95  722 

95  727 

95  732 

95  737 

95  742 

95  746 

95  751 

95  756 

907 

95  761 

95  766  95  770 

95  775 

95  780 

95  785 

95  789 

95  794 

95  799 

95  804 

908 

95  809 

95  813  95  818 

95  823 

95  828 

95  832 

95  837 

95  842 

95  847 

95  852 

909 

95  856 

95  861  95  866 

95  871 

95  875 

95  880 

95  885 

95  890  95  895 

95  899 

910 

95  904 

95  909  95  914 

95  918 

95  923 

95  928 

95  933 

95  938 

95  942 

95  947 

911 

95  952 

95  957  95  961 

95  966 

95  971 

95  976 

95  980 

95  985 

95  990 

95  995 

912 

95  999 

96  004  96  009 

96  014 

96  019 

96  023 

96  028 

96  033 

96  038 

96  042 

913 

96  047 

96  052  96057 

96  061 

96  066 

96  071 

96  076 

96  080 

96  085 

96  090 

914 

96095 

96  099  96104 

96109 

96114 

96118 

96123 

96128 

96133 

96137 

915 

96142 

96147  96152 

96156 

96161 

96166 

96171 

96175 

96180  96185 

916 

96190 

96194  96199 

96  204 

96  209 

96  213 

96  218 

96  223 

96  227 

96  232 

917 

96  237 

96  242  96  246 

96  251 

96  256 

96  261 

96  265 

96  270 

96  275 

96  280 

918 

96  284 

96  289  96  294 

96  298 

96  303 

96  308 

96  313 

96  317 

96  322 

96  327 

919 

96  332 

96  336  96  341 

96  346 

96  350 

96  355 

96  360 

96  365 

96  369 

96  374 

920 

96  379 

96  384  96  388 

96  393 

96  398 

96  402 

96  407 

96  412 

96  417 

96  421 

921 

96426 

96  431  96  435 

96  440 

96  445 

96  450 

96  454 

96  459 

96  464 

96  468 

922 

96  473 

96  478  96483 

96487 

96  492 

96  497 

96  501 

96  506 

96  511 

96  515 

923 

96  520  96  525  96  530 

96  534 

96  539 

96  544 

96  548 

96  553 

96  558 

96  562 

924 

96  567 

96  572  96  577 

96  581 

96  586 

96  591 

96  595 

96  600 

96  605 

96  609 

925 

96  614 

96619  96624 

96628 

96  633 

96  638 

96  642 

96  647 

96  652 

96  656 

926 

96  661 

96  666  96  670 

96  675 

96  680 

96  6S5 

96  6S9 

96  694 

96  699 

96  703 

927 

96  708 

96  713  96  717 

96  722 

96  727 

96  731 

96  736 

96  741 

96  745 

96  750 

928 

96  755 

96  759  96  764 

96  769 

96  774 

96  778 

96  783 

96  788 

96  792 

96  797 

929 

96  802 

96  806  96  811 

96  816 

96  820 

96  825 

96  830 

96  834 

96  839 

96  844 

930 

96  848 

96  853  96  858 

96  862 

96  867 

96  872 

96  876 

96  881 

96  886 

96  890 

931 

96  895 

96  900  96  904 

96  909 

96  914 

96  918 

96  923 

96  928 

96  932 

96  937 

932 

96942 

96  946  96  951 

96  956 

96  960 

96  965 

96  970 

96  974 

96  979 

96  984 

933 

96988 

96993  96  997 

97  002 

97  007 

97  011 

97  016 

97  021 

97  025 

97  030 

934 

97  035 

97  039  97  044 

97  049 

97  053 

97  058 

97  063 

97  067 

97  072 

97Q77 

935 

97  081 

97  086  97  090 

97095 

97100 

97104 

97109 

97114 

97118 

97123 

936 

97128 

97132  97137 

97142 

97  146 

97151 

97  155 

97  160 

97  165 

97169 

937 

97174 

97179  97183 

97188 

97192 

97197 

97  202 

97  206 

97  211 

97  216 

938 

97  220 

97  225  97  230 

97  234 

97  239 

97  243 

97  248 

97  253 

97  257 

97  262 

939 

97  267 

97  271  97  276 

97  280 

97  285 

97  290 

97  294 

97  299 

97  304 

97  308 

940 

97  313 

97  317  97  322 

97  327 

97  331 

97  336 

97  340 

97  345 

97  350 

97  354 

941 

97  359 

97  364  97  368 

97  373 

97  377 

97  3S2 

97  387 

97  391 

97  396 

97  400 

942 

97  405 

97  410  97  414 

97  419 

97  424 

97  428 

97  433 

97  437 

97  442 

97  447 

943 

97  451 

97  456  97  460  97  465 

97  470 

97  474 

97  479 

97  483 

97  488 

97  493 

944 

97  497 

97  502  97  506 

97  511 

97  516 

97  520 

97  525 

97  529 

97  534 

97  539 

945 

97  543 

97  548  97  552 

97  557 

97  562 

97  566 

97  571 

97  575 

97  580  97  585 

946 

97  589 

97  594  97  598 

97  603 

97  607 

97  612 

97  617 

97  621 

97  626 

97  630 

947 

97  635 

97  640  97  644 

97  649 

97  653 

97  658 

97  663 

97  667 

97  672 

97  676 

948 

97  681 

97  685  97  690 

97  695 

97  699 

97  704 

97  708 

97  713 

97  717 

97  722 

949 

97  727 

97  731  97  736 

97  740 

97  745 

97  749 

97  754 

97.759 

97  763 

97  768 

950 

97  772 

97  777  97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

N 

O 

1    2 

3 

4 

5 

6 

7 

8 

9 

900-950 


950-1000 


19 


N 

o 

1 

2    3    4 

5 

6    7    8    9 

950 

97  772 

97  777 

97  782  97  786  97  791 

97  795 

97  800  97  804  97  809  97  813 

951 

97  818 

97  823 

97  827  97  832  97  836 

97  841 

97  845  97  850  97  855  97  859 

952 

97  864 

97  868 

97  873  97  877  97  882 

97  886 

97  891  97  896  97  900  97  905 

953 

97  909 

97  914 

97  918  97  923  97  928 

97  932 

97  937  97  941  97  946  97  950 

954 

97955 

97  959 

97  964  97  968  97  973 

97  978 

97  982  97  987  97  991  97  996 

955 

98  000 

98005 

98  009  98014  98019 

98023 

98  028  98  032  98  037  98  041 

956 

98  046 

98  050 

98  055  98  059  98  064 

98068 

98  073  98  078  98  082  98  087 

957 

98  091 

98  096 

98100  98105  98109 

98114 

98118  98123  98127  98132 

958 

98137 

98141 

98146  98150  98155 

98159 

98164  98168  98173  98177 

959 

98182 

98186 

98191  98195  98  200 

98  204 

98  209  98  214  98  218  98  223 

960 

98  227 

98  232 

98  236  98  241  98  245 

98  250 

98  254  98  259  98  263  98  268 

961 

98  272 

98  277 

98  281  98  286  98  290 

98  295 

98  299  98  304  98  308  98  313 

962 

98  318 

98  322 

98  327  98  331  98  336 

98  340 

98  345  98  349  98  354  98  358 

963 

98  363 

98  367 

98  372  98  376  98  381 

98  385 

98  390  98  394  98  399  98  403 

964 

98  408 

98  412 

98  417  98  421  98  426 

98  430  98435  98439  98  444  98  448 

965 

98453 

98  457 

98462  98  466  98  471 

98  475 

98480  98  484  98  489  98  493 

966 

98  498 

98  502 

98  507  98  511  98  516 

98  520 

98  525  98  529  98  534  98  538 

967 

98  543 

98  547 

98  552  98  556  98  561 

98  565 

98  570  98  574  98  579  98  583 

968 

98  588 

98  592 

98  597  98  601  98  605 

98  610 

98  614  98  619  98  623  98  628 

969 

98  632 

98  637 

98  641  98  646  98650 

98  655 

98659  98  664  98  668  98  673 

970 

98  677 

98  682 

98  686  98  691  98  695 

98  700 

98  704  98  709  98  713  98  717 

971 

98  722 

98  726 

98  731  98  735  98  740 

98  744 

98  749  98  753  98  758  98  762 

972 

98  767 

98  771 

98  776  98  780  98  784 

98  789 

98  793  98  798  98  802  98  807 

973 

98  811 

98  816 

98  820  98  825  98  829 

98  834 

98  838  98  843  98  847  98  851 

974 

98  856  98  860  98  865  98  869  98  874 

98  878 

98  883  98  887  98  892  98  896 

975 

98900  98  905 

98  909  98  914  98  918 

98  923 

98927  98932  98936  98  941 

976 

98  945 

98949 

98  954  98958  98  963 

98  967 

98972  98976  98981  98  985 

977 

98  989 

98994 

98  998  99  003  99  007 

99  012 

99  016  99021  99025  99029 

978 

99  034 

99  038 

99  043  99  047  99  052 

99  056 

99  061  99  065  99  069  99  074 

979 

99078 

99  0S3 

99  087  99  092  99  096 

99100  99105  99109  99114  9^118 

980 

99123 

99127 

99131  99136  99140 

99145 

99149  99154  99158  99162 

981 

99167 

99171 

99176  99180  99185 

99189 

99193  99198  99  202  99  207 

982 

99  211 

99  216 

99  220  99  224  99  229 

99  233 

99  238  99  242  99  247  99  251 

983 

99  255 

99  260 

99  264  99  269  99  273 

99  277 

99  282  99  286  99  291  99  295 

984 

99  300 

99  304 

99  308  99  313  99  317 

99  322 

99  326  99  330  99  335  99  339 

985 

99  344 

99  348 

99  352  99  357  99  361 

99  366 

99  370  99  374  99  379  99383 

986 

99  388 

99  392 

99  396  99  401  99  405 

99  410 

99  414  99  419  99  423  99  427 

987 

99  432 

99  436 

99  441  99  445  99  449 

99  454 

99  458  99  463  99467  99  471 

988 

99  476 

99  480 

99  484  99  489  99  493 

99  498 

99  502  99  506  99  511  99  515 

989 

99  520 

99  524 

99  528  99  533  99  537 

99  542 

99  546  99  550  99  555  99  559 

990 

99  564 

99  568 

99  572  99  577  99  581 

99  585 

99  590  99  594  99  599  99  603 

991 

99  607 

99  612 

99  616  99  621  99  625 

99  629 

99  634  99  638  99  642  99  647 

992 

99  651 

99  656 

99  660  99  664  99  669 

99  673 

99  677  99  682  99  686  99  691 

993 

99  695 

99  699 

99  704  99  708  99  712 

99  717 

99  721  99  726  99  730  99  734 

994 

99  739 

99  743 

99  747  99  752  99  756 

99  760 

99  765  99  769  99  774  99  778 

995 

99  782 

99  787 

99  791  99  795  99  800 

99  804 

99  808  99  813  99  817  99  822 

996 

99  826 

99  830  99  835  99  839  99  843 

99  848 

99  852  99  856  99  861  99  865 

997 

99870 

99  874 

99  878  99  883  99  887 

99  891 

99  896  99  900  99904  99  909 

998 

99913 

99  917 

99  922  99  926  99  930 

99935 

99  939  99944  99  948  99  952 

999 

99957 

99  961 

99  965  99  970  99  974 

99978 

99  983  99  987  99  991  99  996 

lOOO 

00000 

00  004 

00009  00013  00017 

00  022  00026  00030  00  035  00  0*39 

N 

0 

1 

2    3    4 

5 

6    7    8    9 

950-1000 


20 


TABLE 

[I. 

THE  LOGAEITHMS 

OF  THE 

TRIGONOMETRIC 

FUNCTIONS : 

Prom  0°  to  0°  3',  or  89°  57'  to  90°,  for  every  second ; 

From  0°  to  2°,  or  88° 

to  90°,  for 

every  ten  seconds  i 

Prom  1°  t 
Note.     T 

log  sin 

d  89°,  for  every  minut( 

3. 

0  is  to  be  appended. 

log  tan  =  log  sin 
log  cos  =  10. 00  000 

o  all  the  logarithms  —  1 

0° 

tt 
0 

O' 

1' 

2' 

tt 

tt 

O' 

1' 

2' 

tt 



6. 46  373 

6.76476 

60 

30 

6.16270 

6.  63  982 

6.  86  167 

30 

1 

4.  68  557 

6. 47  090 

6.  76  836 

59 

31 

6. 17  694 

6. 64  462 

6.  86  455 

29 

2 

4. 98  660 

6.  47  797 

6.  77  193 

58 

32 

6. 19  072 

6.  64  936 

6.  86  742 

28 

3 

5. 16  270 

6.  48  492 

6.  77  548 

57 

33 

6.  20  409 

6.  65  406 

6.  87  027 

27 

4 

5.  28  763 

6. 49  175 

6.  77  900 

56 

34 

6.  21  705 

6. 65  870 

6.87  310 

26 

5 

5.38  454 

6.  49  849 

6.  7S  248 

55 

35 

6.  22  964 

6.66330 

6.  87  591 

25 

6 

5.  46  373 

6.50  512 

6.  78  595 

54 

36 

6.24188 

6.  66  785 

6.  87  870 

24 

7 

5.  53  067 

6.  51 165 

6.  78  938 

53 

37 

6.  25  378 

6.  67  235 

6.  88  147 

23 

8 

5.  58  866 

6.  51  808 

6.  79  278 

52 

38 

6.  26  536 

6.  67  680 

6.  88  423 

22 

9 

5. 63  982 

6.  52  442 

6.  79  616 

51 

39 

6.  27  664 

6.  68  121 

6.88  697 

21 

10 

5.68  557 

6.  53  067 

6.  79  952 

50 

40 

6.  28  763 

6.  68  557 

6.88  969 

20 

11 

5.  72  697 

6.  53  683 

6.  80  285 

49 

41 

6.  29  836 

6.  68  990 

6.  89  240 

19 

12 

5.  76  476 

6.  54  291 

6.80  615 

48 

42 

6.  30  8S2 

6.69  418 

6.  89  509 

18 

13 

5.79  952 

6\  54  890 

6.  80  943 

47 

43 

6.  31  904 

6.  69  841 

6.  89  776 

17 

14 

5.  83  170 

6.  55  481 

6. 81  268 

46 

44 

6.  32  903 

6.  70  261 

6.90  042 

16 

15 

5.  86  167 

6.  56  064 

6.  81  591 

45 

45 

6.  33  879 

6.70  676 

6.  90  306 

15 

16 

5.  88  969 

6.  56  639 

6.  81  911 

44 

46 

6.  34  833 

6.  71  088 

6.  90  568 

14 

17 

5.  91  602 

6.  57  207 

6.  82  230 

43 

47 

6.  35  767 

6.  71  496 

6.  90  829 

13 

18 

5.  94  085 

6.  57  767 

6.  82  545 

42 

48 

6.  36  682 

6.  71  900 

6. 91  088 

12 

19 

5.  96  433 

6.  58  320 

6.  82  859 

41 

49 

6.37  577 

6.  72  300 

6.  91  346 

11 

20 

5.98  660 

6.  58  866 

6.  83  170 

40 

50 

6.  38  454 

6.  72  697 

6. 91  602 

io 

21 

6.  00  779 

6.  59  406 

6.  83  479 

39 

51 

6.39  315 

6.  73  090 

6.  91  857 

9 

22 

6.  02  800 

6.  59  939 

6. 83  786 

38 

52 

6.  40  158 

6.  73  479 

6.  92  110 

8 

23 

6. 04  730 

6.  60  465 

6.  84  091 

37 

53 

6.  40  9S5 

6.  73  865 

6. 92  362 

7 

24 

6. 06  579 

6.  60  985 

6.  84  394 

36 

54 

6.  41  797 

6.  74  248 

6.  92  612 

6 

25 

6.08  351 

6.  61  499 

6.  84  694 

35 

55 

6. 42  594 

6.  74  627 

6. 92  861 

5 

26 

6. 10  055 

6.  62  007 

6.  84  993 

34 

56 

6.  43  376 

6.  75  003 

6.  93  109 

4 

27 

6. 11  694 

,  6.  62  509 

6. 85  289 

33 

57 

6. 44  145 

6.  75  376 

6. 93  355 

3 

28 

6. 13  2731/  6. 63  006 

6.  85  584 

32 

58 

6.  44  900 

6.  75  746 

6. 93  599 

2 

29 

6. 14  797 

6.  63  496 

6.  85  876 

31 

59 

6. 45  643 

6.  76  112 

6. 93  843 

1 

30 

tt 

6. 16  270 

6.  63  982 

6.  86  167 

30 

60 

6.46  373 

6.76476 

6.  94  085 

O 

59' 

58' 

57' 

tt 

tt 

59' 

58' 

57' 

tt 

log  cot  =  log  COS 
log  sin  =  10. 00  000 


89c 


log  cos 


\,  I  ft 


0° 

21 

/  " 

log  sin 

log  tan 

log  COS 

tt   t 

t  tt 

log  sin 

7.  46  373 

log  tan 

log  COS 

"  / 

O  0 





10.00000 

0  6O 

1O0 

7. 46  373 

10.00000 

0  50 

10 

5. 68  557 

5.  68  557 

]  0.00000 

50 

10 

7.  47  090 

7. 47  091 

10.00000 

50 

20 

5.98  660 

5. 98  660 

10.00000 

40 

20 

7. 47  797 

7.  47  797 

10.00000 

40 

30 

6. 16  270 

6. 16  270 

10.00000 

30 

30 

7. 48  491 

7.  48  492 

10.00000 

30 

40 

6.  28  763 

6.  28  763 

10.00000 

20 

40 

7. 49  175 

7. 49  176 

10.00000 

20 

50 

6.  38  454 

6.  38  454 

10.00000 

10 

50 

7. 49  849 

7. 49  849 

10.00000 

10 

1  0 

6. 46  373 

6.46  373 

10.00000 

0  59 

110 

7.50  512 

7.50  512 

10.00000 

049 

10 

6.  53  067 

6.  53  067 

10.00000 

50 

10 

7.  51 165 

7.  51 165 

10.00000 

50 

20 

6.  58  866 

6.  58  866 

10.00000 

40 

20 

7.  51  808 

7.  51  809 

10.00000 

40 

x  30 

6.  63  982 

6. 63  982 

10.00000 

30 

30 

7.  52  442 

7.  52  443 

10.00000 

30 

40 

6.  68  557 

6.  68  557 

10.00000 

20 

40 

7.  53  067 

7.  53  067 

10.0CC00 

20 

50 

6.  72  697 

6.  72  697 

10.00000 

10 

50 

7.  53  683 

7.  53  683 

10.00000 

10 

2  0 

6.76476 

6.76  476 

10.00000 

0  58 

12  0 

7.  54  291 

7.  54  291 

1000000 

048 

10 

6.  79  952 

6.  79  952 

10.00000 

50 

10 

7.  54  890 

7.  54  890 

10.00COO 

50 

20 

6.  83  170 

6.  83  170 

10.00000 

40 

20 

7.  55  481 

7.  55  481 

10.COOOO 

40 

30 

6.  86  167 

6.  86  167 

10.00000 

30 

30. 

7.  56  064 

7.  56  064 

10.00000 

30 

40 

6.  88  969 

6.  88  969 

10.00000 

20 

40 

7.  56  639 

7.  56  639 

10.00000 

20 

50 

6.  91  602 

6.  91  602 

10.00000 

10 

50 

7.  57  206, 

7.57  207 

10.00000 

10 

3  0 

6.  94  0S5 

6.94  085 

10.00000 

057 

13  0 

7.  57  767* 

7.  57  767 

10.00000 

047 

10 

6.  96  433 

6.  96  433 

10.00000 

50 

10 

7.  58  320 

7.  58  320 

10.00000 

50 

20 

6.  98  660 

6.  98  661 

10.00000 

40 

20 

7.  58  866 

7.  58  867 

1000000 

40 

30 

7. 00  779 

7. 00  779 

10.00000 

30 

30 

7.59406 

7.  59  406 

10.00000 

30 

40 

7. 02  800 

7.  02  800 

10.00000 

20 

40 

7.59939 

7.  59  939 

10.00000 

20 

50 

7.04  730 

7.  04  730 

10.00000 

10 

50 

7.60465 

7.60466 

10.00000 

10 

4  0 

7. 06  579 

7. 06  579 

10.00000 

0  56 

14  0 

7.  60  985 

7.  60  986 

10.00000 

046 

10 

7.08  351 

7.08  352 

10.00000 

50 

10 

7.  61  499 

7.  61  500 

10.00000 

50 

20 

7.10  055 

7.10055 

10.00000 

40 

20 

7.  62  007 

7.  62  008 

10.00000 

40 

30 

7. 11  694 

7. 11  694 

10.00000 

30 

30 

7. 62  509 

7.62  510 

10.00000 

30 

40 

7. 13  273 

7. 13  273 

10.00000 

20 

40 

7.  63  006 

7. 63  006 

10.00000 

20 

50 

7. 14  797 

7. 14  797 

10.00000 

10 

50 

7. 63  496 

7. 63  497 

10.00000 

10 

5  0 

7. 16  270 

7. 16  270 

10.00000 

0  55 

15  0 

7. 63  982 

7. 63  982 

10.00000 

045 

10 

7. 17  694 

7. 17  694 

10.00000 

50 

10 

7.64461 

7.  64  462 

10.00000 

50 

20 

7. 19  072 

7.19  073 

10.00000 

40 

20 

7.  64  936 

7.64  937 

10.00000 

40 

30 

7.  20  409 

7.20409 

10.00000 

30 

30 

7. 65  406 

7.  65  406 

10.00000 

30 

40 

7.  21  705 

7. 21  705 

10.00000 

20 

40 

7. 65  870 

7.  65  871 

10.00000 

20 

50 

7.  22  964 

7.  22  964 

10.00000 

10 

50 

7. 66  330 

7. 66  330 

10.00000 

10 

6  0 

7.  24  188 

7.  24  188 

10.00000 

0  54 

16  0 

7. 66  784 

7.  66  785 

10.00000 

0  44 

10 

7.  25  378 

7.  25  378 

10.00000 

50 

10 

7.  67  235 

7. 67  235 

10.00000 

50 

20 

7.  26  536 

7.26536 

10.00000 

40 

20 

7.  67  680 

7. 67  680 

10.00000 

40 

30 

7.  27  664 

7.  27  664 

10.00000 

30 

30 

7.  68  121 

7. 68  121 

10.00000 

30 

40 

7.  28  763 

7.  28  764 

10.00000 

20 

40 

7.  68  557 

7.  68  558 

9.99999 

20 

50 

7. 29  836 

7.  29  836 

10.00000 

10 

50 

7. 68  989 

7.68  990 

9.99999 

10 

7  0 

7.  30  882 

7.  30  882 

10.00000 

0  53 

170 

7.69417 

7.69418 

9.  99  999 

0  43 

10 

7.  31  904 

7. 31  904 

10.00000 

50 

10 

7.  69  841 

7.  69  842 

9.  99  999 

50 

20 

7.  32  903 

7.  32  903 

10.00000 

40 

20 

7.  70  261 

7.  70  261 

9.99  999 

40 

30 

7.  33  879 

7.  33  879 

10.00000 

30 

30 

7.70676 

7.  70  677 

9.  99  999 

30 

40 

7. 34  833 

7. 34  833 

10.00000 

20 

40 

7.  71  088 

7.  71  088 

9.99  999 

20 

50 

7. 35  767 

7. 35  767 

10.00000 

10 

50 

7.  71  496 

7.  71 496 

9.99999 

10 

8  0 

7. 36  682 

7. 36  682 

10.00000 

0  52 

18  0 

7.71900 

7.  71  900 

9.99  999 

042 

10 

7.37  577 

7.37  577 

10.00000 

50 

10 

7.  72  300 

7.  72  301 

9.99999 

50 

20 

7. 38  454 

7.38455 

10.00000 

40 

20 

7.  72  697 

7.  72  697 

9. 99  999 

40 

30 

7.39  314 

7.39  315 

10.00000 

30 

30 

7.  73  090 

7.  73  090 

9.99999 

30 

40 

7. 40  158 

7.  40 158 

10.00000 

20 

40 

7.  73  479 

7.  73  480 

9.99  999 

20 

50 

7.40985 

7.40985 

10.00000 

10 

50 

7.  73  865 

7.  73  866 

9.99999 

10 

9  0 

7. 41  797 

7. 41  797 

10.00000 

0  51 

19  0 

7.  74  248 

7.  74  248 

9.99999 

0  41 

10 

7. 42  594 

7. 42  594 

10.00000 

50 

10 

7.  74  627 

7.  74  628 

9.99  999 

50 

20 

7. 43  376 

7. 43  376 

10.00000 

40 

20 

7.  75  003 

7.75  004 

9.99  999 

40 

30 

7. 44  145 

7. 44  145 

10.00000 

30 

30 

7.  75  376 

7.  75  377 

9.99  999 

30 

40 

7.  44  900 

7.44  900 

10.00000 

20 

40 

7.  75  745 

7.  75  746 

9.99  999 

20 

50 

7. 45  643 

7. 45  643 

10.00000 

10 

50 

7.76112 

7.  76  113 

9.99  999 

10 

1O0 

7.46373 

7.  46  373 

10.00000 

0  50 

20  0 

7.76  475 

7.76476 

9.99  999 

040 

t  tt 

log  cos 

log  cot 

log  sin 

it   t 

t  ft 

log  cos 

log  cot 

log  sin 

tt  t 

m 


22 

0° 

/  " 

log  sin 

log  tan 

7.76  476 

log  COS 

9. 99  999 

//  t 

t  tt 

log  sin 

log  tan 

log  cos 

9.  99  998 

tt  t 

200 

7.76475 

040 

30  0 

7. 94  084 

7. 94  086 

0  30 

10 

7.  76  836 

7.  76  837 

9.  99  999 

50 

10 

7.  94  325 

7. 94  326 

9. 99  998 

50 

20 

7.  77  193 

7.  77  194 

9.  99  999 

40 

20 

7. 94  564 

7.  94  566 

9.  99  998 

40 

30 

7.  77  548 

7.  77  549 

9. 99  999 

30 

30 

7.  94  802 

7.  94  804 

9. 99  998 

30 

40 

7.  77  899 

7.  77  900 

9. 99  999 

20 

40 

7. 95  039 

7.  95  040 

9.  99  998 

20 

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99  520 

00  480 

85  187 

19 

42 

83  940 

98  029 

01971 

85  912 

18 

42 

84  720 

99  545 

00  455 

85  175 

18 

43 

83  954 

98  054 

01946 

85  900 

17 

43 

84  733 

99  570 

00  430 

85  162 

17 

44 

83  967 

98  079 

01921 

85  888 

16 

44 

84  745 

99  596 

00  404 

85  150 

16 

45 

83  980 

98104 

01896 

85  876 

15 

45 

84  758 

99  621 

00  379 

85  137 

15 

46 

83  993 

98130 

01870 

85  864 

14 

46 

84  771 

99  646 

00  354 

85  125 

14 

47 

84  006 

98155 

01845 

85  851 

13 

47 

84  784 

99  672 

00  328 

85  112 

13 

48 

84  020 

98  180 

01820 

85  839 

12 

48 

84  796 

99  697 

00  303 

85  100 

12 

49 

84  033 

98  206 

01794 

85  827 

11 

49 

84  809 

99  722 

00  278 

85  087 

11 

SO 

84  046 

98  231 

01769 

85  815 

IO 

50 

84  822 

99  747 

00  253 

85  074 

IO 

51 

84  059 

98  256 

01744 

85  803 

9 

51 

84  835 

99  773 

00  227 

85  062 

9 

52 

84  072 

98  281 

01719 

85  791 

8 

52 

84  847 

99  798 

00  202 

85  049 

8 

53 

84  085 

98  307 

01693 

85  779 

7 

53 

84  860 

99  823 

00177 

85  037 

7 

54 

84  098 

98  332 

01668 

85  766 

6 

54 

81873 

99  848 

00152 

85  024 

6 

55 

84112 

98  357 

01643 

85  754 

5 

55 

84  885 

99  874 

00126 

85  012 

5 

56 

84125 

98  383 

01617 

85  742 

4 

56 

84  898 

99  899 

00101 

84  999 

4 

57 

84138 

98  408 

01592 

85  730 

3 

57 

84  911 

99  924 

00  076 

84  986 

3 

58 

84  151 

98  433 

01567 

85  718 

2 

58 

84  923 

99  949 

00  051 

84  974 

2 

59 

84164 

98  458 

01542 

85  706 

1 

59 

84  936 

99  975 

00  025 

84  961 

1 

60 

84177 

98  484 

01516 

85  693 

O 

60 

84  949 

00  000 

00  000 

84  949 

O 

9 

log  COS 

9 

log  cot 

IO 

log  tan 

9 

log  sin 

9 

log  COS 

IO 

log  cot 

IO 

log  tan 

9 

t 

t 

r 

log  sin 

t 

46c 


45 


49 


TABLE   III. 

For  Determining  with  Greater  Accuracy  than  can  be  done  by 

Means  of  Table  II: 

1.    log  sin,  log  tan,  and  log  cot,  when  the  angle  is  between  0°  and  2° ; 

2.    log  cos,  log  tan,  and  log  cot,  when  the  angle  is  between  88°  and  90°  ; 

3.    The  value  of  the  angle  when  the  logarithm  of  the  function  does  not  lie 

between  the  limits  8.  54  684  and  11.  45  316. 

FORMULAS  FOR   THE   USE   OF  THE   NUMBERS^   AND  T 

when  the  angle  x  is  between  0°  and  2° : 

log  sin  x  =  log  x  +  S. 

log  tan  x  =  log  x  +  T. 

—*— 

Values  of  S  and  T. 

X 

S 

log  sin  x 

X 

T 

log  tan  x 

X 

T 

log  tan  x 

0 

0 

__ 

5  146 

8.  39  713 

4.  68  557 

4.  68  557 

4.  68  567 

2  409 

4,. 68  556 

8.  06  740 

200 

4.  68  558 

6.  98  660 

5  424 

4.  68  568 

8.  41  999 

3  417 

4.  68  555 

8.  21  920 

1  726 

4.  68  559 

7.  92  263 

5  689 

4.  68  569 

8.  44  072 

3  823 

4.  68  555 

8.  26  795 

2  432 

4.  68  560 

8.  07  156 

5  941 

4.  68  570 

8.  45  955 

4  190 

4.  68  554 

8.  30  776 

2  976 

4.  68  561 

8.  15  924 

6  184 

4.  68  571 

8.  47  697 

4  840 

4.  68  553 

8.  37  038 

3  434 

4.  68  562 

8.  22  142 

6  417 

4.  68  572 

8.  49  305 

5  414 

4.  6S  552 

8.  41  904 

3  838 

4.  68  563 

8.  26  973 

6  642 

4.  68  573 

8.  50  802 

5  932 

4.  68  551 

8.  45  872 

4  204 

4.  68  564 

8.  30  930 

6  859 

4.  68  574 

8.  52  200 

6  408 

4.  68  550 

8.  49  223 

4  540 

4.  68  565 

8.  34  270 

7  070 

4.  68  575 

8.  53  516 

6  633 

4.  68  550 

8.50.721 

4  699 

4.  68  565 

8.  35  766 

7  173 

4.  68  575 

8.  54  145 

6  851 

4.  68  549 

8.  52  125 

4  853 

4.  68  566 

8.  37  167 

7  274 

8.  54  753 

7  267 

8.  54  684 

5  146 

8.  39  713 

X 

S 

log  sin  x 

X 

T 

log  tan  x 

X 

T 

log  tan  x 

50 


TABLE   IV.  — NATURAL    FUNCTIONS. 


/ 

o° 

1° 

2° 

3° 

4° 

r 

sin 

cos 

sin  cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

0000 

1.000 

0175  9998 

0349 

9994 

0523 

9986 

0698 

9976 

60 

1 

0003 

1.000 

0177  9998 

0352 

9994 

0526 

9986 

0700 

9975 

59 

2 

0006 

1.000 

0180  9998 

0355 

9994 

0529 

9986 

0703 

9975 

58 

3 

0009 

1.000 

0183  999S 

0358 

9994 

0532 

9986 

0706 

9975 

57 

4 

0012 

1.000 

0186  9998 

0361 

9993 

0535 

9986 

0709 

9975 

56 

5 

0015 

1.000 

0189  9998 

0364 

9993 

0538 

9986 

0712 

9975 

55 

6 

0017 

1.000 

0192  9998 

0366 

9993 

0541 

9985 

0715 

9974 

54 

7 

0020 

1.000 

0195  9998 

0369 

9993 

0544 

9985 

0718 

9974 

53 

8 

0023 

1.000 

0198  9998 

0372 

9993 

0547 

9985 

0721 

9974 

52 

9 

0026 

1.000 

0201  9998 

0375 

9993 

0550 

9985 

0724 

9974 

51 

io 

0029 

1.000 

0204  9998 

0378 

9993 

0552 

99S5 

0727 

9974 

50 

11 

0032 

1.000 

0207  9998 

03S1 

9993 

0555 

9985 

0729 

9973 

49 

12 

0035 

1.000 

0209  9998 

03S4 

9993 

0558 

9984 

0732 

9973 

48 

13 

0038 

1.000 

0212  9998 

0387 

9993 

0561 

9984 

0735 

9973 

47 

14 

0041 

1.000 

0215  9998 

.0390 

9992 

0564 

9984 

0738 

9973 

46 

15 

0044 

1.000 

0218  9998 

0393 

9992 

0567 

9984 

0741 

9973 

45 

16 

0047 

1.000 

0221  9998 

0396 

9992 

0570 

9984 

0744 

9972 

44 

17 

0049 

1.000 

0224  9997 

0398 

9992 

0573 

9984 

0747 

9972 

43 

18 

0052 

1.000 

0227  9997 

0401 

9992 

0576 

9983 

0750 

9972 

42 

19 

0055 

1.000 

0230  9997 

0404 

9992 

0579 

9983 

0753 

9972 

41 

20 

0058 

1.000 

0233  9997 

0407 

9992 

0581 

9983 

0756 

9971 

40 

21 

0061 

1.000 

0236  9997 

0410 

9992 

05S4 

9983 

0758 

9971 

39 

22 

0064 

1.000 

0239  9997 

0413 

9991 

0587 

9983 

0761 

9971 

38 

23 

0067 

i.ooo 

0241  9997 

0416 

9991 

0590 

9983 

0764 

9971 

37 

24 

0070 

1.000 

0244  9997 

0419 

9991 

0593 

9982 

0767 

9971 

36 

25 

0073 

1.000 

0247  9997 

0422 

9991 

0596 

99S2 

0770 

9970 

35 

26 

0076 

1.000 

0250  9997 

0425 

9991 

0599 

9982 

0773 

9970 

34 

27 

0079 

1.000 

0253  9997 

0427 

9991 

0602 

9982 

0776 

9970 

33 

28 

0081 

1.000 

0256  9997 

0430 

9991 

0605 

99S2 

0779 

9970 

32 

29 

0084 

1.000 

0259  9997 

0433 

9991 

0608 

9982 

0782 

9969 

31 

30 

0087 

1.000 

0262  9997 

0436 

9990 

0610 

9981 

07S5 

9969 

30 

31 

0090 

1.000 

0265  9996 

0439 

9990 

0613 

9981 

0787 

9969 

29 

32 

0093 

1.000 

0268  9996 

0442 

9990 

0616 

99S1 

0790 

9969 

28 

33 

0096 

1.000 

0270  9996 

0445 

9990 

0619 

9981 

0793 

9968 

27 

34 

0099 

1.000 

0273  9996 

0448 

9990 

0622 

9981 

0796 

9968 

26 

35 

0102 

9999 

0276  9996 

0451 

9990 

0625 

99S0 

0799 

9968 

25 

36 

0105 

9999 

0279  9996 

0-454 

9990 

0628 

9980 

0S02 

9968 

24 

37 

0108 

9999 

0282  9996 

0457 

9990 

0631 

9980 

0805 

9968 

23 

38 

0111 

9999 

0285  9996 

0459 

9989 

0634 

9980 

0S08 

9967 

22 

39 

0113 

9999 

0288  9996 

0462 

9989 

0637 

9980 

0811 

9967 

21 

40 

0116 

9999 

0291  9996 

0465 

99S9 

0640 

9980 

0814 

9967 

20 

41 

0119 

9999 

0294  9996 

0468 

9989 

0642 

9979 

0816 

9967 

19 

42 

0122 

9999 

0297  9996 

0471 

9989 

0645 

9979 

0819 

9966 

IS 

43 

0125 

9999 

0300  9996 

0474 

99S9 

0648 

9979 

0822 

9966 

17 

44 

0128 

9999 

0302  9995 

0477 

9989 

0651 

9979 

0825 

9966 

16 

45 

0131 

9999 

0305  9995 

0480 

99S8 

0654 

9979 

0828 

9966 

15 

46 

0134 

9999 

0308  9995 

0483 

9988 

0657 

9978 

0831 

9965 

14 

47 

0137 

9999 

0311  9995 

0486 

99S8 

0660 

9978 

0834 

9965 

13 

48 

0140 

9999 

0314  9995 

0488 

9988 

0663 

9978 

0837 

9965 

12 

49 

0143 

9999 

0317  9995 

0491 

9988 

0666 

9978 

0840 

9965 

11 

50 

0145 

9999 

0320  9995 

0494 

9988 

0669 

997S 

0843 

9964 

10 

51 

0148 

9999 

0323  9995 

0497 

9988 

0671 

9977 

0845 

9964 

9 

52 

0151 

9999 

0326  9995 

0500 

99S7 

'  0674 

9977 

0848 

9964 

8 

53 

0154 

9999 

0329  9995 

0503 

9987 

0677 

9977 

0851 

9964 

7 

54 

0157 

9999 

0332  9995 

0506 

9987 

06S0 

9977 

OSS  4 

9963 

6 

55 

0160 

9999 

0334  9994 

0509 

9987 

0683 

9977 

0857 

9963 

5 

56 

0163 

9999 

0337  9994 

0512 

9987 

0686 

9976 

0S60 

9963 

4 

57 

0166 

9999 

0340  9994 

0515 

9987 

0689 

9976 

0863 

9963 

3 

58 

0169 

9999 

0343  9994 

0518 

9987 

0692 

9976 

0866 

9962 

2 

59 

0172 

9999 

0346  9994 

0520 

9986 

0695 

9976 

0869 

9962 

1 

60 

0175 

9999 

0349  9994 

0523 

9986 

0698 

9976 

0872 

9962 

O 

cos   sin 
89° 

cos  sin 
88° 

cos  sin 

87° 

cos  sin 
86° 

cos   sin 

85° 

f 

r 

NATURAL   SINES    AND   COSINES.  51 


/ 

5° 

6° 

7° 

8° 

9° 

f 

sin   cos 

sin 

cos 

sin 

cos 

sin  cos 

sin  cos 

o 

0872  9962 

1045 

9945 

1219 

9925 

1392  9903 

1564  9877 

60 

1 

0874  9962 

1048 

9945 

1222 

9925 

1395  9902 

1567  9876 

59 

2 

0877  9961 

1051 

9945 

1224 

9925 

1397  9902 

1570  9876 

58 

3 

0880  9961 

1054 

9944 

1227 

9924 

1400  9901 

1573  9876 

57 

4 

0883  9961 

1057 

9944 

1230 

9924 

1403  9901 

1576  9875 

56 

5 

0886  9961 

1060 

9944 

1233 

9924 

1406  9901 

1579  9875 

55 

6 

0889  9960 

1063 

9943 

1236 

9923 

1409  9900 

1582  9874 

54 

7 

0892  9960 

1066 

9943 

1239 

9923 

1412  9900 

1584  9874 

53 

.  8 

0895  9960 

1068 

9943 

1241 

9923 

1415  9899 

1587  9873 

52 

9 

0898  9960 

1071 

9942 

1245 

9922 

1418  9899 

1590  9873 

51 

io 

0901  9959 

1074 

9942 

1248 

9922 

1421  9899 

1593  9872 

50 

11 

0903  9959 

1077 

9942 

1250 

9922 

1423  9898 

1596  9872 

49 

12 

0906  9959 

1080 

9942 

1253 

9921 

1426  9898 

1599  9871 

48 

13 

0909  9959 

1083 

9941 

1256 

9921 

1429  9897 

1602  9871 

47 

14 

0912  9958 

10S6 

9941 

1259 

9920 

1432  9897 

1605  9870 

46 

15 

0915  9958 

1089 

9941 

1262 

9920 

1435  9897 

1607  9870 

45 

16 

0918  9958 

1092 

9940 

1265 

9920 

1438  9896 

1610  9869 

44 

17 

0921  9958 

1094 

9940 

1268 

9919 

1441  9896 

1613  9869 

43 

18 

0924  9957 

1097 

9940 

1271 

9919 

1444  9895 

1616  9869 

42 

19 

0927  9957 

1100 

9939 

1274 

9919 

1446  9895 

1619  9868 

41 

20 

0929  9957 

1103 

9939 

1276 

9918 

1449  9894 

1622  9868 

40 

21 

0932  9956 

1106 

9939 

1279 

9918 

1452  9894 

1625  9867 

39 

22 

0935  9956 

1109 

9938 

1282 

9917 

1455  9894 

1628  9867 

38 

23 

0938  9956 

1112 

9938 

1285 

9917 

1458  9893 

1630  9866 

37. 

24 

0941  9956 

1115 

9938 

1288 

9917 

1461  9893 

1633  9866 

36 

25 

0944  9955 

1118 

9937 

1291 

9916 

1464  9892 

1636  9865 

35 

26 

0947  9955 

1120 

9937 

1294 

9916 

1467  9892 

1639  9865 

34 

27 

0950  9955 

1123 

9937 

1297 

9916 

1469  9891 

1642  9864 

33 

28 

0953  9955 

1126 

9936 

1299 

9915 

1472  9891 

1645  9864 

32 

29 

0956  9954 

1129 

9936 

1302 

9915 

1475  9891 

1648  9863 

31 

30 

0958  9954 

1132 

9936 

1305 

9914 

1478  9890 

1650  9863 

30 

31 

0961  9954 

1135 

9935 

1308 

9914 

1481  9890 

1653  9862 

29 

32 

0964  9953 

1138 

9935 

1311 

9914 

1484  9889 

1656  9862 

28 

33 

0967  9953 

1141 

9935 

1314 

9913 

1487  9889 

1659  9861 

27 

34 

0970  9953 

1144 

9934 

1317 

9913 

1490  9888 

1662  9861 

26 

35 

0973  9953 

1146 

9934 

1320 

9913 

1492  9888 

1665  9860 

25 

36 

0976  9952 

1149 

9934 

1323 

>9912 

1495  9888 

1668  9S60 

24 

37 

0979  9952 

1152 

9933 

1325 

9912 

1498  9887 

1671  9859 

23 

38 

0982  9952 

1155 

9933 

1328 

9911 

1501  9887 

1673  9859 

22 

39 

0985  9951 

1158 

9933 

1331 

9911 

1504  9886 

1676  9859 

21 

40 

0987  9951 

1161 

9932 

1334 

9911 

1507  9886 

1679  9858 

20 

41 

0990  9951 

1164 

9932 

1337 

9910 

1510  9885 

1682  9858 

19 

42 

0993  9951 

1167 

9932 

1340 

9910 

1513  9885 

1685  9857 

18 

43 

0996  9950 

1170 

9931 

1343 

9909 

1515  9884 

1688  9857 

17 

44 

0999  9950 

1172 

9931 

1346 

9909 

1518  9884 

1691  9856 

16 

45 

1002  9950 

1175 

9931 

1349 

9909 

1521  9884 

1693  9856 

15 

46 

1005  9949 

1178 

9930 

1351 

9908 

1524  9883 

1696  9855 

14 

47 

1008  9949 

1181 

9930 

1354 

9908 

1527  9883 

1699  9855 

13 

48 

1011  9949 

1184 

9930 

1357 

9907 

1530  9882 

1702  9854 

12 

49 

1013  9949 

1187 

9929 

1360 

9907 

1533  9882 

1705  9854 

11 

50 

1016  9948 

1190 

9929 

1363 

9907 

1536  9881 

1708  9853 

IO 

51 

1019  9948 

1193 

9929 

1366 

9906 

1538  9881 

1711  9853 

9 

52 

1022  9948 

1196 

9928 

1369 

9906 

1541  9880 

1714  9852 

8 

53 

1025  9947 

1198 

9928 

1372 

9905 

1544  9880 

1716  9852 

7 

54 

1028  9947 

1201 

9928 

1374 

9905 

1547  9880 

1719  9851 

6 

55 

1031  9947 

1204 

9927 

1377 

9905 

1550  9879 

1722  9851 

5 

56 

1034  9946 

1207 

9927 

1380 

9904 

1553  9879 

1725  9850 

4 

57 

1037  9946 

1210 

9927 

1383 

9904 

1556  9878 

1728  9850 

3 

58 

1039  9946 

1213 

9926 

1386 

9903 

1559  9878 

1731  9849 

2 

59 

1042  9946 

1216 

9926 

1389 

9903 

1561  9877 

1734  9S49 

1 

60 

1045  9945 

1219 

9925 

1392 

9903 

1564  9877 

1736  9848 

O 

cos  sin 
84° 

cos  sin 
83° 

cos  sin 
82° 

cos  sin 
81° 

cos  sin 

f 

80° 

f 

52 


NATUKAL   SINES    AND   COSINES. 


' 

io° 

11° 

12° 

13° 

14° 

r 

sin  cos 

sin 

cos 

sin  cos 

sin 

cos 

sin 

cos 

o 

1736  9848 

1908 

9816 

2079  9781 

2250 

9744 

2419 

9703 

60 

1 

1739  9848 

1911 

9816 

2082  9781 

2252 

9743 

2422 

9702 

59 

2 

1742  9847 

1914 

9815 

2085  9780 

2255 

9742 

2425 

9702 

58 

3 

1745  9847 

1917 

9815 

2088  9780 

2258 

9742 

2428 

9701 

57 

4 

1748  9846 

1920 

9814 

2090  9779 

2261 

9741 

2431 

9700 

56 

5 

1751  9846 

1922 

9813 

2093  9778 

2264 

9740 

2433 

9699 

55 

6 

1754  9845 

1925 

9813 

2096  9778 

2267 

9740 

2436 

9699 

54 

7 

1757  9845 

1928 

9812 

2099  9777 

2269 

9739 

2439 

9698 

53 

8 

1759  9844 

1931 

9812 

2102  9777 

2272 

9738 

2442 

9697 

52 

9 

1762  9843 

1934 

9811 

2105  9776 

2275 

9738 

2445 

9697 

51 

io 

1765  9843 

1937 

9811 

2108  9775 

2278 

9737 

2447 

9696 

50 

11 

1768  9842 

1939 

9810 

2110  9775 

22S1 

9736 

2450 

9695 

49 

12 

1771  9842 

1942 

9810 

2113  9774 

2284 

9736 

2453 

9694 

48 

13 

1774  9841 

1945 

9809 

2116  9774 

2286 

9735 

2456 

9694 

47 

14 

1777  9841 

1948 

9808 

2119  9773 

2289 

9734 

2459 

9693 

46 

15 

1779  9840 

1951 

9808 

2122  9772 

2292 

9734 

2462 

9692 

45 

16 

1782  9840 

1954 

9807 

2125  9772 

2295 

9733 

2464 

9692 

44 

17 

1785  9839 

1957 

9807 

2127  9771 

2298 

9732 

2467 

9691 

43 

18 

1788  9839 

1959 

9806 

2130  9770 

2300 

9732 

2470 

9690 

42 

19 

1791  9838 

1962 

9806 

2133  9770 

2303 

9731 

2473 

9689 

41 

20 

1794  9838 

1965 

9805 

2136  9769 

2306 

9730 

2476 

9689 

40 

21 

1797  9837 

1968 

9804 

2139  9769 

2309 

9730 

2478 

9688 

39 

22 

1799  9837 

1971 

9804 

2142  9768 

2312 

9729 

2481 

9687 

38 

23 

1802  9836 

1974 

9803 

2145  9767 

2315 

9728 

2484 

9687 

37 

24 

1805.9836 

1977 

9803 

2147  9767 

2317 

9728 

2487 

9686 

36 

25 

1808  9835 

1979 

9802 

2150  9766 

2320 

9727 

2490 

9685 

35 

26 

1811  9835 

1982 

9802 

2153  9765 

2323 

9726 

2493 

9684 

34 

27 

1814  9834 

1985 

9801 

2156  9765 

2326 

9726 

249>  9684 

33 

28 

1817  9834 

1988 

9800 

2159  9764 

2329 

9725 

2498 

9683 

32 

29 

1819  9833 

1991 

9800 

2162  9764 

2332 

9724 

2501 

9682 

31 

30 

1822  9833 

1994 

9799 

2164  9763 

2334 

9724 

2504 

9681 

30 

31 

1825  9832 

1997 

9799 

2167  9762 

2337 

9723 

2507 

9681 

29 

32 

1828  9831 

1999 

9798 

2170  9762 

2340 

9722 

2509 

9680 

28 

33 

1831  9831 

2002 

9798 

2173  9761 

2343 

9722 

2512 

9679 

27 

34 

1834  9830 

2005 

9797 

2176  9760 

2346 

9721 

2515 

9679 

26 

35 

1837  9830 

2008 

9796 

2179  9760 

2349 

9720 

2518 

9678 

25 

36 

1840  9829 

2011 

9796 

2181  9759 

2351 

9720 

2521 

9677 

24 

37 

1842  9829 

2014 

9795 

2184  9759 

2354 

9719 

2524 

9676 

23 

38 

1845  9828 

2016 

9795 

2187  9758 

2357 

9718 

2526 

9676 

22 

39 

1848  9828 

2019 

9794 

2190  9757 

2360 

9718 

2529 

9675 

21 

40 

1851  9827 

2022 

9793 

2193  9757 

2363 

9717 

2532 

9674 

20 

41 

1854  9827 

2025 

9793 

2196  9756 

2366 

9716 

2535 

9673 

19 

42 

1857  9826 

2028 

9792 

2198  9755 

2368 

9715 

2538 

9673 

18 

43 

1860  9826 

2031 

9792 

2201  9755 

2371 

9715 

2540 

9672 

17 

44 

1862  9825 

2034 

9791 

2204  9754 

2374 

9714 

2543 

9671 

16 

45 

1865  9825 

2036 

9790 

2207  9753 

2377 

9713 

2546 

9670 

15 

46 

1868  9824 

2039 

9790 

2210  9753 

2380 

9713 

2549 

9670 

14 

47 

1871  9823 

2042 

9789 

2213  9752 

2383 

9712 

2552 

9669 

13 

48 

1874  9823 

2045 

9789 

2215  9751 

2385 

9711 

2554 

9668 

12 

49 

1877  9822 

2048 

9788 

2218  9751 

2388 

9711 

2557 

9667 

11 

50 

1880  9822 

2051 

9787 

2221  9750 

2391 

9710 

2560 

9667 

10 

51 

1882  9821 

2054 

9787 

2224  9750 

2394 

9709 

2563 

9666 

9 

52 

1885  9821 

2056 

9786 

2227  9749 

2397 

9709 

2566 

9665 

8 

53 

1888  9820 

2059 

9786 

2230  9748 

2399 

9708 

2569 

9665 

7 

54 

1891  9820 

2062 

9785 

2233  9748 

2402 

9707 

2571 

9664 

6 

55 

1894  9819 

2065 

9784 

2235  9747 

2405 

9706 

2574 

9663 

5 

56 

1897  9818 

2068 

9784 

2238  9746 

2408 

9706 

2577 

9662 

4 

57 

1900  9S18 

2071 

9783 

2241  9746 

2411 

9705 

2580 

9662 

3 

58 

1902  9817 

2073 

9783 

2244  9745 

2414 

9704 

2583 

9661 

2 

59 

1905  9817 

2076 

9782 

2247  9744 

2416 

9704 

2585 

9660 

1 

60 

1908  9816 

2079 

9781 

2250  9744 

2419 

9703 

2588 

9659 

O 

cos  sin 
79° 

cos  sin 
78° 

cos  sin 

77° 

cos  sin 
76° 

cos   sin 

75° 

f 

r 

NATURAL    SINES    AND    COSINES. 


53 


/ 

15° 

16° 

17° 
sin  cos 

18° 

19° 

t 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

2588 

9659 

2756 

9613 

2924  9563 

3090 

9511 

3256 

9455 

60 

1 

2591 

9659 

2759 

9612 

2926  9562 

3093 

9510 

3258 

9454 

59 

2 

2594 

9658 

2762 

9611 

2929  9561 

3096 

9509 

3261 

9453 

58 

3 

2597 

9657 

2165 

9610 

2932  9560 

3098 

9508 

3264 

9452 

57 

4 

2599 

9656 

2768 

9609 

2935  9560 

3101 

9507 

3267 

9451 

56 

5 

2602 

9655 

2770 

9609 

2938  9559 

3104 

9506 

3269 

9450 

55 

6 

2605 

9655 

2773 

9608 

2940  9558 

3107 

9505 

3272 

9449 

54 

7 

2608 

9654 

2776 

9607 

2943  9557 

3110 

9504 

3275 

9449 

53 

8 

2611 

9653 

2779 

9606 

2946  9556 

3112 

9503 

3278 

9448 

52 

9 

2613 

9652 

2782 

9605 

2949  9555 

3115 

9502 

32S0 

9447 

51 

io 

2616 

9652 

2784 

9605 

2952  9555 

3118 

9502 

3283 

9446 

50 

11 

2619 

9651 

2787 

9604 

2954  9554 

3121 

9501 

3286 

9445 

49 

12 

2622 

9650 

2790 

9603 

2957  9553 

3123 

9500 

3289 

9444 

48 

13 

2625 

9649 

2793 

9602 

2960  9552 

3126 

9499 

3291 

9443 

47 

14 

2628 

9649 

2795 

9601 

2963  9551 

3129 

9498 

3294 

9442 

46 

15 

2630 

9648 

2798 

9600 

2965  9550 

3132 

9497 

3297 

9441 

45 

16 

2633 

9647 

2801 

9600 

2968  9549 

3134 

9496 

3300 

9440 

44 

17 

2636 

9646 

2804 

9599 

2971  9548 

3137 

9495 

3302 

9439 

43 

18 

2639 

9646 

2807 

9598 

2974  9548 

3140 

9494 

3305 

9438 

42 

19 

2642 

9645 

2809 

9597 

2977  9547 

3143 

9493 

3308 

9437 

41 

20 

2644 

9644 

2812 

9596 

2979  9546 

3145 

9492 

3311 

9436 

40 

21 

2647 

9643 

2815 

9596 

2982  9545 

3148 

9492 

3313 

9435 

39 

22 

2650 

9642 

2818 

9595 

2985  9544 

3151 

9491 

3316 

9434 

38 

23 

2653 

9642 

2821 

9594 

2988  9543 

3154 

9490 

3319 

9433 

37 

24 

2656 

9641 

2823 

9593 

2990  9542 

3156 

9489 

3322 

9432 

36 

25 

2658 

9640 

2826 

9592 

2993  9542 

3159 

9488 

3324 

9431 

35 

26 

2661 

9639 

2829 

9591 

2996  9541 

3162 

9487 

3327 

9430 

34 

27 

2664 

9639 

2832 

9591 

2999  9540 

3165 

9486 

3330 

9429 

33 

28 

2667 

9638 

2835 

9590 

3002  9539 

3168 

9485 

3333 

9428 

32 

29 

2670 

9637 

2837 

9589 

3004  9538 

3170 

9484 

3335 

9427 

31 

30 

2672 

9636 

2840 

9588 

3007  9537 

3173 

9483 

3338 

9426 

30 

31 

2675 

9636 

2843 

9587 

3010  9536 

3176 

9482 

3341 

9425 

29 

32 

2678 

9635 

2846 

9587 

3013.  9535 

3179 

9481 

3344 

9424 

28 

33 

2681 

9634 

2849 

9586 

301^  9535 
3018*  9534 

3181 

9480 

3346 

9423 

27 

34 

2684 

9633 

2851 

9585 

3184 

9480 

3349 

9423 

26 

35 

26S6 

9632 

2S54 

9584 

3021  9533 

3187 

9479 

3352 

9422 

25 

36 

2689 

9632 

2857 

9583 

3024  9532 

3190 

9478 

3355 

9421 

24 

37 

2692 

9631 

2860 

9582 

3026  9531 

3192 

9477 

3357 

9420 

23 

38 

2695 

9630 

2862 

9582 

3029  9530 

3195 

9476 

3360 

9419 

22 

39 

2698 

9629 

2865 

9581 

3032  9529 

3198 

9475 

3363 

9418 

21 

40 

2700 

9628 

2868 

9580 

3035  9528 

3201 

9474 

3365 

9417 

20 

41 

2703 

9628 

2871 

9579 

3038  9527 

3203 

9473 

3368 

9416 

19 

42 

2706 

9627 

2874 

9578 

3040  9527 

3206 

9472 

3371 

9415 

18 

43 

2709 

9626 

2876 

9577 

3043  9526. 

3209 

9471 

3374 

9414 

17 

44 

2712 

9625 

2879 

9577 

3046  9525 

3212 

9470 

3376 

9413 

16 

45 

2714 

9625 

2882 

9576 

3049  9524 

3214 

9469 

3379 

9412 

15 

46 

2717 

9624 

2885 

9575 

3051  9523 

3217 

9468 

3382 

9411 

14 

47 

2720 

9623 

2888 

9574 

3054  9522 

3220 

9467 

3385 

9410 

13 

48 

2723 

9622 

2890 

9573 

3057  9521 

3223 

9466 

3387 

9409 

12 

49 

2726 

9621 

2893 

9572 

3060  9520 

3225 

9466 

3390 

9408 

11 

50 

2728 

9621 

2896 

9572 

3062  9520 

3228 

9465 

3393 

9407 

IO 

51 

2731 

9620 

2899 

9571 

3065  9519 

3231 

9464 

3396 

9406 

9 

52 

2734 

9619 

2901 

9570 

3068  9518 

3234 

9463 

3398 

9405 

8 

53 

2737 

9618 

2904 

9569 

3071  9517 

3236 

9462 

3401 

9404 

7 

54 

2740 

9617 

2907 

9568 

3074  9516 

3239 

9461 

3404 

9403 

6 

55 

2742 

9617 

2910 

9567 

3076  9515 

3242 

9460 

3407 

9402 

5 

56 

2745 

9616 

2913 

9566 

3079  9514 

3245 

9459 

3409 

9401 

4 

57 

2748 

9615 

2915 

9566 

3082  9513 

3247 

9458 

3412 

9400 

3 

58 

2751 

9614 

2918 

9565 

3085  9512 

3250 

9457 

3415 

9399 

2 

59 

2754 

9613 

2921 

9564 

3087  9511 

3253 

9456 

3417 

9398 

1 

60 

2756 

9613 

2924 

9563 

3090  9511 

3256 

9455 

3420 

9397 

O 

cos  sin 

74° 

cos  sin 
73° 

cos  sin 

72° 

cos  sin 

71° 

cos   sin 

70° 

f 

f 

54 

NATURAL 

SINES  AND 

COSINES. 

/ 

20° 

21° 

22° 

23° 

sin  cos 

24° 

t 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

0 

3420 

9397 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

60 

1 

3423 

9396 

3586 

9335 

3749 

9271 

3910 

9204 

4070 

9134 

59 

2 

3426 

9395 

3589 

9334 

3751 

9270 

3913 

9203 

4073 

9133 

58 

3 

3428 

9394 

3592 

9333 

3754 

9269 

3915 

9202 

4075 

9132 

57 

4 

3431 

9393 

3595 

9332 

3757 

9267 

3918 

9200 

4078 

9131 

56. 

5 

3434 

9392 

3597 

9331 

3760 

9266 

3921 

9199 

4081 

9130 

55 

6 

3437 

9391 

3600 

9330 

3762 

9265 

3923 

9198 

4083 

9128 

54 

7 

3439 

9390 

3603 

9328 

3765 

9264 

3926 

9197 

4086 

9127 

53 

8 

3442 

9389 

3605 

9327 

3768 

9263 

3929 

9196 

4089 

9126 

52 

9 

3445 

9388 

3608 

9326 

3770 

9262 

3931 

9195 

4091 

9125 

51 

10 

3448 

9387 

3611 

9325 

3773 

9261 

3934 

9194 

4094 

9124 

50 

11 

3450 

9386 

3614 

9324 

3776 

9260 

3937 

9192 

4097 

9122 

49 

12 

3453 

9385 

3616 

9323 

3778 

9259 

3939 

9191 

4099 

9121 

48 

13 

3456 

9384 

3619 

9322 

3781 

9258 

3942 

9190 

4102 

9120 

47 

14 

3458 

9383 

3622 

9321 

3784 

9257 

3945 

9189 

4105 

9119 

46 

15 

3461 

9382 

3624 

9320 

3786 

9255 

3947 

9188 

4107 

9118 

45 

16 

3464 

9381 

3627 

9319 

3789 

9254 

3950 

9187 

4110 

9116 

44 

17 

3467 

9380 

3630 

9318 

3792 

9253 

3953 

9186 

4112 

9115 

43 

18 

3469 

9379 

3633 

9317 

3795 

9252 

3955 

9184 

4115 

9114 

42 

19 

3472 

9378 

3635 

9316 

3797 

9251 

3958 

9183 

4118 

9113 

41 

20 

3475 

9377 

3638 

9315 

3800 

9250 

3961 

9182 

4120 

9112 

40 

21 

3478 

9376 

3641 

9314 

3803 

9249 

3963 

9181 

4123 

9110  • 

39 

22 

3480 

9375 

3643 

9313 

3805 

9248 

3966 

9180 

4126 

9109 

38 

23 

3483 

9374 

3646 

9312 

3808 

9247 

3969 

9179 

4128 

9108 

37 

24 

3486 

9373 

3649 

9311 

3811 

9245 

3971 

9178 

4131 

9107 

36 

25 

3488 

9372 

3651 

9309 

3813 

9244 

3974 

9176 

4134 

9106 

35 

26 

3491 

9371 

3654 

9308 

3816 

9243 

3977 

9175 

4136 

9104 

34 

27 

3494 

9370 

3657 

9307 

3819 

9242 

3979 

9174 

4139 

9103 

33 

28 

3497 

9369 

3660 

9306 

3821 

9241 

3982 

9173 

4142 

9102 

32 

29 

3499 

9368 

3662 

9305 

3824 

9240 

39S5 

9172 

4144 

9101 

31 

30 

3502 

9367 

.  3665 

9304 

3827 

9239 

3987 

9171 

4147 

9100 

30 

31 

3505 

9366 

3668 

9303 

3830 

9238 

3990 

9169 

4150 

9098 

29 

32 

3508 

9365 

3670 

9302 

3832 

9237 

3993 

9168 

4152 

9097 

28 

33 

3510 

9364 

3673 

9301 

3835 

9235 

3995 

9167 

4155 

9096 

27 

34 

3513 

9363 

3676 

9300 

3838 

9234 

3998 

9166 

4158 

9095 

26 

35 

3516 

9362 

3679 

9299 

3840 

9233 

4001 

9165 

4160 

9094 

25 

36 

3518 

9361 

3681 

9298 

3843 

9232 

4003 

9164 

4163 

9092 

24 

37 

3521 

9360 

3684 

9297 

3846 

9231 

4006 

9162 

4165 

9091 

23 

38 

3524 

9359 

3687 

9296 

3848 

9230 

4009 

9161 

4168 

9090 

22 

39 

3527 

9358 

3689 

9295 

3851 

9229 

4011 

9160 

4171 

9088 

21 

40 

3529 

9356 

3692 

9293 

3854 

9228 

4014 

9159 

4173 

9088 

20 

41 

3532 

9355 

3695 

9292 

3856 

9227 

4017 

9158 

4176 

9086 

19 

42 

3535 

9354 

3697 

9291 

3859 

9225 

4019 

9157 

4179 

9085 

18 

43 

3537 

9353 

3700 

9290 

3862 

9224 

4022 

9155 

4181 

9084 

17 

44 

3540 

9352 

3703 

9289 

3864 

9223 

4025 

9154 

41S4 

9083 

16 

45 

3543 

9351 

3706 

9288 

3867 

9222 

4027 

9153 

4187 

9081 

15 

46 

3546 

9350 

3708 

9287 

3870 

9221 

4030 

9152 

4189 

9080 

14 

47 

3548 

9349 

3711 

9286 

3872 

9220 

4033 

9151 

4192 

9079 

13 

48 

3551 

9348 

3714 

9285 

3875 

9219 

4035 

9150 

4195 

9078 

12 

49 

3554 

9347 

3716 

9284 

3878 

9218 

4038 

9148 

4197 

9077 

11 

50 

3557 

9346 

3719 

9283 

3881 

9216 

4041 

9147 

4200 

9075 

io 

51 

3559 

9345 

3722 

9282 

3883 

9215 

4043 

9146 

4202 

9074 

9 

52 

3562 

9344 

3724 

9281 

3886 

9214 

4046 

9145 

4205 

9073 

8 

53 

3565 

9343 

3727 

9279 

3889 

9213 

4049 

9144 

4208 

9072 

7 

54 

3567 

9342 

3730 

9278 

3891 

9212 

4051 

9143 

4210 

9070 

6 

55 

3570 

9341 

3733 

9277 

3894 

9211 

4054 

9141 

4213 

9069 

5 

56 

3573 

9340 

3735 

9276 

3897 

9210 

4057 

9140 

4216 

9068 

4 

57 

3576 

9339 

3738 

9275 

3899 

9208 

4059 

9139 

4218 

9067 

3 

58 

3578 

9338 

3741 

9274 

3902 

9207 

4062 

9138 

4221 

9066 

2 

59 

3581 

9337 

3743 

9273 

3905 

9206 

4065 

9137 

4224 

9064 

1 

60 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

4226 

9063 

O 

cos   sin 
69° 

cos  sin 
68° 

cos  sin 
67° 

cos  sin 
66° 

cos  sin 
65° 

f 

f 

NATURAL   SINES   AND   COSINES. 


55 


/ 

25° 

26° 

27° 

28° 

29° 

t 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

4226 

9Q63 

4384 

8988 

4540 

8910 

4695 

8829 

4848 

8746 

60 

1 

4229 

9062 

4386 

8987 

4542 

8909 

4697 

8828 

4851 

8745 

59 

2 

4231 

9061 

4389 

8985 

4545 

8907 

4700 

8827 

4853 

8743 

58 

3 

4234 

9059 

4392 

8984 

4548 

8906 

4702 

8825 

4856 

8742 

57 

4 

4237 

9058 

4394 

8983 

4550 

8905 

4705 

8824 

4858 

8741 

56 

5 

4239 

9057 

4397 

8982 

4553 

8903 

4708 

8823 

4861 

8739 

55 

6 

4242 

9056 

4399 

8980 

4555 

8902 

4710 

8821 

4863 

8738 

54 

7 

4245 

9054 

4402 

8979 

4558 

8901 

4713 

8820 

4866 

8736 

53 

8 

4247 

9053 

4405 

8978 

4561 

8899 

4715 

8819 

4868 

8735 

52 

9 

4250 

9052 

4407 

8976 

4563 

8898 

4718 

8817 

4871 

8733 

51 

io 

4253 

9051 

4410 

8975 

4566 

8897 

4720 

8816 

4874 

8732 

SO 

11 

4255 

9050 

4412 

8974 

4568 

8895 

4723 

8814 

4876 

8731 

49 

12 

4258 

9048 

4415 

8973 

4571 

8894 

4726 

8813 

4879 

8729 

48 

13 

4260 

9047 

4418 

8971 

4574 

8893 

4728 

8812 

4881 

8728 

47 

14 

4263 

9046 

4420 

8970 

4576 

8892 

4731 

8810 

4884 

8726 

46 

15 

4266 

9045 

4423 

8969 

4579 

8890 

4733 

8809 

4886 

8725 

45 

16 

4268 

9043 

4425 

8967 

4581 

8889 

4736 

8808 

4889 

8724- 

44 

17 

4271 

9042 

4428 

8966 

4584 

8888 

4738 

8806 

4891 

8722 

43 

18 

4274 

9041 

4431 

8965 

4586 

8886 

4741 

8805 

4894 

8721 

42 

19 

4276 

9040 

4433 

8964 

4589 

8885 

4743 

8803 

4896 

8719 

41 

20 

4279 

9038 

4436 

8962 

4592 

8884 

4746 

8802 

4899 

8718 

40 

21 

4281 

9037 

4439 

8961 

4594 

8882 

4749 

8801 

4901 

8716 

39 

22 

4284 

9036 

4441 

8960 

4597 

8881 

4751 

8799 

4904 

8715 

38 

23 

4287 

9035 

4444 

8958 

4599 

8879 

4754 

8798 

4907 

8714 

37 

24 

4289 

9033 

4446 

8957 

4602 

8878 

4756 

8796 

4909 

8712 

36 

25 

4292 

9032 

4449 

8956 

4605 

8877 

4759 

8795 

4912 

8711 

35 

26 

4295 

9031 

4452 

8955 

4607 

8875 

4761 

8794 

4914 

8709 

34 

27 

4297 

9030 

4454 

8953 

4610 

8874 

4764 

8792 

4917 

8708 

33" 

28 

4300 

9028 

4457 

8952 

4612 

8873 

4766 

8791 

4919 

8706 

32 

29 

4302 

9027 

4459 

8951 

4615 

8871 

4769 

8790 

4922 

8705 

31 

30 

4305 

9026 

4462 

8949 

4617 

8870 

4772 

8788 

4924 

8704 

30 

31 

4308 

9025 

4465 

8948 

4620 

8869 

4774 

8787 

4927 

8702 

29 

32 

4310 

9023 

4467 

8947 

4623 

8867 

4777 

8785 

4929 

8701 

28 

33 

4313 

9022 

4470 

8945 

4625 

8866 

4779 

8784 

4932 

8699 

27 

34 

4316 

9021 

4472 

8944 

4628 

8865 

4782 

8783 

4934 

8698 

26 

35 

4318 

9020 

4475 

8943 

4630 

8863 

4784 

8781 

4937 

8696 

25 

36 

4321 

9018 

4478 

8942 

4633 

8862 

4787 

8780 

4939 

8695 

24 

37 

4323 

9017 

4480 

8940 

4636 

8861 

4789 

8778 

4942 

8694 

23 

38 

4326 

9016 

4483 

8939 

4638 

8859 

4792 

8777 

4944 

8692 

22 

39 

4329 

9015 

4485 

8938 

4641 

8858 

4795 

8776 

4947 

8691 

21 

40 

4331 

9013 

4488 

8936 

4643 

8857 

4797 

8774 

4950 

8689 

20 

41 

4334 

9012 

4491 

8935 

4646 

8855 

4800 

8773 

4952 

8688 

19 

42 

4337 

9011 

4493 

8934 

4648 

8854 

4802 

8771 

4955 

8686 

18 

43 

4339 

9010 

4496 

8932 

4651 

8853 

4805 

8770 

4957 

8685 

17 

44 

4342 

9008 

4498 

8931 

4654 

8851 

4807 

8769' 

4960 

8683 

16 

45 

4344 

9007 

4501 

8930 

4656 

8850 

4810 

8767 

4962 

8682 

15 

46 

4347 

9006 

4504 

8928 

4659 

8849 

4812 

8766 

4965 

8681 

14 

47 

4350 

9004 

4506 

8927 

4661 

8S47 

4815 

8764 

4967 

8679 

13 

*& 

4352 

9003 

4509 

8926 

4664 

8846 

4818 

8763 

4970 

8678 

12 

49 

4355 

9002 

4511 

8925 

4666 

8844 

4820 

8762 

4972 

8676 

11 

50 

4358 

9001 

4514 

8923 

4669 

8843 

4823 

8760 

4975 

8675 

10 

51 

4360 

8999 

4517 

8922 

4672 

8842 

4825 

8759 

4977 

8673 

9 

52 

4363 

8998 

4519 

8921 

4674 

8840 

4828 

8757 

4980 

8672 

8 

53 

4365 

8997 

4522 

8919 

4677 

8839 

4830 

8756 

4982 

8670 

7 

54 

4368 

8996 

4524 

8918 

4679 

8838 

4833 

8755 

4985 

8669 

6 

55 

,  4371 

8994 

4527 

8917 

4682 

8836 

4835 

8753 

4987 

8668 

5 

56 

4373 

8993 

4530 

8915 

4684 

8835 

4838 

8752 

4990 

8666 

4 

57 

4376 

8992 

4532 

8914 

4687 

8834 

4840 

8750 

4992 

8665 

3 

58 

4378 

8990 

4535 

8913 

4690 

8832 

4843 

8749 

4995 

8663 

2 

59 

km 

8989 

4537 

8911 

4692 

8831 

4846 

8748 

4997 

8662 

1 

60 

4384 

8988 

4540 

8910 

4695 

8829 

4848 

8746 

5000 

8660 

O 

cos 

sin 

cos  sin 
63° 

cos  sin 
62° 

cos 

sin 

cos   sin 
60° 

f 

64° 

61° 

f 

56 


NATURAL   SINES    AND    COSINES. 


y 

30° 

31° 

32° 

sin  cos 

33° 

sin  cos 

34° 

t 

sin 

cos 

sin 

cos 

sin 

cos 

O 

5000 

8660 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

60 

1 

5003 

8659 

5153 

8570 

5302 

8479 

5449 

8385 

5594 

8289 

59 

2 

5005 

8657 

5155 

8569 

5304 

8477 

5451 

8384 

5597 

8287 

58 

3 

5008 

8656 

5158 

8567 

5307 

8476 

5454 

8382 

5599 

8285 

57 

4 

5010 

8654 

5160 

8566 

5309 

8474 

5456 

8380 

5602 

8284 

56 

5 

5013 

8653 

5163 

8564 

5312 

8473 

5459 

8379 

5604 

8282 

55 

6 

5015 

8652 

5165 

8563 

5314 

8471 

5461 

8377 

5606 

8281 

54 

7 

5018 

8650 

5168 

8561 

5316 

8470 

5463 

8376 

5609 

8279 

53 

8 

5020 

8649 

5170 

8560 

5319 

8468 

5466 

8374 

5611 

8277 

52 

9 

5023 

8647 

5173 

8558 

5321 

8467 

5468 

8372 

5614 

8276 

51 

io 

5025 

8646 

5175 

8557 

5324 

8465 

5471 

8371 

5616 

8274 

50 

11 

5028 

8644 

5178 

8555 

5326 

8463 

5473 

8369 

5618 

8272 

49 

12 

5030 

8643 

5180 

8554 

5329 

8462 

5476 

8368 

5621 

8271 

48 

13 

5033 

8641 

5183 

8552 

5331 

8460 

5478 

8366 

5623 

8269 

47 

14 

5035 

8640 

5185 

8551 

5334 

8459 

5480 

8364 

5626 

8268 

46 

15 

5038 

8638 

5188 

8549 

5336 

8457 

5483 

8363 

5628 

8266 

45 

16 

5040 

8637 

5190 

8548 

5339 

8456 

5485 

8361 

5630 

8264 

44 

17 

5043 

8635 

5193 

8546 

5341 

8454 

5488 

8360 

5633 

8263 

43 

18 

5045 

8634 

5195 

8545 

5344 

8453 

5490 

8358 

5635 

8261 

42 

19 

5048 

8632 

5198 

8543 

5346 

8451 

5493 

8356 

5638 

8259 

41 

20 

5050 

8631 

5200 

8542 

5348 

8450 

5495 

8355 

5640 

8258 

40 

21 

5053 

8630 

5203 

8540 

5351 

8448 

5498 

8353 

5642 

8256 

39 

22 

5055 

8628 

5205 

8539 

5353 

8446 

5500 

8352 

5645 

8254 

38 

23 

5058 

8627 

5208 

8537 

5356 

8445 

5502 

8350 

5647 

8253 

37 

24 

5060 

8625 

5210 

8536 

5358 

8443 

5505 

8348 

5650 

8251 

36 

25 

5063 

8624 

5213 

8534 

5361 

8442 

5507 

8347 

5652 

8249 

35 

26 

5065 

8622 

5215 

8532 

5363 

8440 

5510 

8345 

5654 

8248 

34 

27 

5068 

8621 

5218 

8531 

5366 

8439 

5512 

8344 

5657 

8246 

33 

28 

5070 

8619 

5220 

8529 

5368 

8437 

5515 

8342 

5659 

8245 

32 

29 

5073 

8618 

5223 

8528 

5371 

8435 

5517 

8340 

5662 

8243 

31 

30 

5075 

8616 

5225 

8526 

5373 

8434 

5519 

8339 

5664 

8241 

30 

31 

5078 

8615 

5227 

8525 

5375 

8432 

5522 

8337 

5666 

8240 

29 

32 

5080 

8613 

5230 

8523 

5378 

8431 

5524 

8336 

5669 

8238 

28 

33 

5083 

8612 

5232 

8522 

5380 

8429 

5527 

8334 

5671 

8236 

27 

34 

5085 

8610 

5235 

8520 

5383 

8428 

5529 

8332 

5674 

8235 

26 

35 

5088 

8609 

5237 

8519 

5385 

8426 

5531 

8331 

5676 

8233 

25 

36 

5090 

8607 

5240 

8517 

5388 

8425 

5534 

8329 

5678 

8231 

24 

37 

5093 

8606 

5242 

8516 

5390 

8423 

5536 

8328 

5681 

8230 

23 

38 

5095 

8604 

5245 

8514 

5393 

8421 

5539 

8326 

5683 

8228 

22 

39 

5098 

8603 

5247 

8513 

5395 

8420 

5541 

8324 

5686 

8226 

21 

40 

5100 

8601 

5250 

8511 

5398 

8418 

5544 

8323 

5688 

8225 

20 

41 

5103 

8600 

5252 

8510 

5400 

8417 

5546 

8321 

5690 

8223 

19 

42 

5105 

8599 

5255 

8508 

5402 

8415 

5548 

8320 

5693 

8221 

18 

43 

5108 

8597 

5257 

8507 

5405 

8414 

5551 

8318 

5695 

8220 

17 

44 

5110 

8596 

5260 

8505 

5407 

8412 

5553 

8316 

5698 

8218 

16 

45 

5113 

8594 

5262 

8504 

5410 

8410 

5556 

8315 

5700 

8216 

15 

46 

5115 

8593 

5265 

8502 

5412 

8409 

5558 

8313 

5702 

8215 

14 

47 

5118 

8591 

5267 

8500 

5415 

8407 

5561 

8311 

5705 

8213 

13 

48 

5120 

8590 

5270 

8499 

5417 

8406 

5563 

8310 

5707 

8211 

12 

49 

5123 

8588 

5272 

8497 

5420 

8404 

5565 

8308 

5710 

8210 

11 

50 

5125 

8587 

5275 

8496 

5422 

8403 

5568 

8307 

5712 

8208 

IO 

51 

5128 

8585 

5277 

8494 

5424 

8401 

5570 

8305 

5714 

8207 

9 

52 

5130 

8584 

5279 

8493 

5427 

8399 

5573 

8303 

5717 

8205 

8 

53 

5133 

8582 

5282 

8491 

5429 

8398 

5575 

8302 

5719 

8203 

7 

54 

5135 

8581 

5284 

8490 

5432 

8396 

5577 

8300 

5721 

8202 

6 

55 

5138 

8579 

5287 

8488 

5434 

8395 

5580 

8299 

5724 

8200 

5 

56 

5140 

8578 

5289 

8487 

5437 

8393 

5582 

8297 

5726 

8198 

4 

57 

5143 

8576 

5292 

8485 

5439 

8391 

5585 

8295 

5729 

8197 

3 

58 

5145 

8575 

5294 

8484 

5442 

8390 

5587 

8294 

5731 

8195 

2 

59 

5148 

8573 

5297 

8482 

5444 

8388 

5590 

8292 

5733 

8193 

1 

60 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

5736 

8192 

0 

cos  sin 
59° 

cos  sin 
58° 

cos  sin 

57° 

cos  sin 
56° 

cos  sin 
55° 

f 

; 

V 


f 


NATURAL   SINES    AND   COSINES. 


57 


/ 

35° 

36° 

37° 

38° 
sin  cos 

39° 

r 

sin   cos 

sin  cos 

sin 

cos 

sin 

cos 

o 

5736  8192 

5878  8090 

6018 

7986 

6157 

7880 

6293 

7771 

60 

1 

5738  8190 

5880  8088 

6020 

7985 

6159 

7878 

6295 

7770 

59 

2 

5741  8188 

5883  8087 

6023 

7983 

6161 

7877 

6298 

7768 

58 

3 

5743  8187 

5885  8085 

6025 

7981 

6163 

7875 

6300 

7766 

57 

4 

5745  8185 

5887  8083 

6027 

7979 

6166 

7873 

6302 

7764 

56 

5 

5748  8183 

5890  8082 

6030 

7978 

6168 

7871 

6305 

7762 

55 

6 

5750  8181 

5892  8080 

6032 

7976 

6170 

7869 

6307 

7760 

54 

7 

5752  8180 

5894  8078 

6034 

7974 

6173 

7868 

6309 

7759 

53 

8 

5755  8178 

5897  8076 

6037 

7972 

6175 

7866 

6311 

7757 

52 

9 

5757  8176 

5899  8075 

6039 

7971 

6177 

7864 

6314 

7755 

51 

io 

5760  8175 

5901  8073 

6041 

7969 

6180 

7862 

6316 

7753 

50 

11 

5762  8173 

5904  8071 

6044 

7967 

6182 

7860 

6318 

7751 

49 

12 

5764  8171 

5906  8070 

6046 

7965 

6184 

7859 

6320 

7749 

48 

13 

5767  8170 

5908  8068 

6048 

7964 

6186 

7857 

6323 

7748 

47 

14 

5769  8168 

5911  8066 

6051 

7962 

6189 

7855 

6325 

7746 

46 

15 

5771  8166 

5913  8064 

6053 

7960 

6191 

7853 

6327 

7744 

45 

16 

5774  8165 

5915  8063 

6055 

7958 

6193 

7851 

6329 

7742 

44 

17 

5776  8163 

5918  8061 

6058 

7956 

6196 

7850 

6332 

7740 

43 

18 

5779  8161 

5920  8059 

6060 

7955 

6198 

7848 

6334 

7738 

42 

19 

5781  8160 

5922  8058 

6062 

7953 

6200 

7346 

6336 

7737 

41 

20 

5783  8158 

5925  8056 

6065 

7951 

6202 

7844 

6338 

7735 

40 

21 

5786  8156 

5927  8054 

6067 

7950 

6205 

7842 

6341 

7733 

39 

22 

5788  8155 

5930  8052 

6069 

7948 

6207 

7841 

6343 

7731 

38 

23 

5790  8153 

5932  8051 

6071 

7946 

6209 

7839 

6345 

7729 

37 

24 

5793  8151 

5934  8049 

6074 

7944 

6211 

7837 

6347 

7727 

36 

25 

5795  8150 

5937  8047 

6076 

7942 

6214 

7835 

6350 

7725 

35 

26 

5798  8148 

5939  8045 

6078 

7941 

6216 

7833 

6352 

7724 

34 

27 

5800  8146 

5941  8044 

6081 

7939 

6218 

7832 

6354 

7722 

33 

28 

5802  8145 

5944  8042 

6083 

7937 

6221 

7830 

6356 

7720 

32 

29 

5805  8143 

5946  8040 

6085 

7935 

6223 

7828 

6359 

7718 

31 

30 

5807  8141 

5948  8039 

6088 

7934 

6225 

7826 

6361 

7716 

30 

31 

5809  8139 

5951  8037 

6090 

7932 

6227 

7824 

6363 

7714 

29 

32 

5812  8138 

5953  8035 

6092 

7930 

6230 

7822 

6365 

7713 

28 

33 

5814  8136 

5955  8033 

6095 

7928 

6232 

7821 

6368 

7711 

27 

34 

5816  8134 

5958  8032 

6097 

7926 

6234 

7819 

6370 

7709 

26 

35 

5819  8133 

5960  8030 

6099 

7925 

6237 

7817 

6372 

7707 

25 

36 

5821  8131 

5962  8028 

6101 

7923 

6239 

7815 

6374 

7705 

24 

37 

5824  8129 

$965  8026 

6104 

7921 

6241 

7813 

9376 

7703 

23 

38 

5826  8128 

5967  8025 

6106 

7919 

6243 

7812 

6379 

7701 

22 

39 

5828  8126 

5969  8023 

6108 

7918 

6246 

7810 

^6381 

7700 

21 

40 

5831  8124 

5972  8021 

6111 

7916 

6248 

7808 

6383 

7698 

20 

41 

5833  8123 

5974  8020 

6113 

7914 

6250 

7806 

6385 

7696 

19 

42 

5835  8121 

5976  8018 

6115 

7912 

6252 

7804 

6388 

7694 

18 

43 

5838  8119 

5979  8016 

6118 

7910 

6255 

7802 

6390 

7692 

17 

44 

5840  8117 

5981  8014 

6120 

7909 

6257 

7801 

6392 

7690 

16 

45 

5842  8116 

5983  8013 

6122 

7907 

6259 

7799 

6394 

7688 

15 

46 

5845  8114 

5986  8011 

6124 

7905 

6262 

7797 

6397 

7687 

14 

47 

5847  8112 

5988  8009 

6127 

7903 

6264 

7795 

6399 

7685 

13 

48 

5850  8111 

5990  8007 

6129 

7902 

6266 

7793 

6401 

7683 

12 

49 

5852  8109 

5993  8006 

6131 

7900 

6268 

7792 

6403 

7681 

11 

50 

5854  8107 

5995  8004 

6134 

7898 

6271 

7790 

6406 

7679 

IO 

51 

5857  8106 

5997  8002 

6136 

7896 

6273 

7788 

6408 

7677 

9 

52 

5859  8104 

6000  8000 

6138 

7894 

6275 

7786 

6410 

7675 

8 

53 

5861  8102 

6002  7999 

6141 

7893 

6277 

7784 

6412 

7674 

7 

54 

5864  8100 

6004  7997 

6143 

7891 

6280 

7782 

6414 

7672 

6 

55 

5866  8099 

6007  7995 

6145 

7889 

6282 

7781 

6417 

7670 

5 

56 

5868  8097 

6009  7993 

6147 

7887 

6284 

7779 

6419 

7668 

4 

57 

5871  8095 

6011  7992 

6150 

7885 

6286 

7777 

6421 

7666 

3 

58 

5873  8094 

6014  7990 

6152 

7884 

6289 

7775 

6423 

7664 

2 

59 

5875  8092 

6016  7988 

6154 

7882 

6291 

7773 

6426 

7662 

1 

60 

5878  8090 

6018  7986 

6157 

7880 

6293 

7771 

6428 

7660 

O 

cos  sin 
54° 

cos  sin 
53° 

cos  sin 

52° 

cos  sin 
51° 

cos  sin 
50° 

f 

t 

D8 

NAT 

URAL 

SINES  AND 

COSINES. 

/ 

40° 

41c 

42° 
sin  cos 

43° 

44( 

f 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

O 

6428 

7660 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

60 

1 

6430 

7659 

6563 

7545 

6693 

7430 

6822 

7312 

6949 

7191 

59 

2 

6432 

7657 

6565 

7543 

6696 

7428 

6824 

7310 

6951 

7189 

58 

3 

6435 

7655 

6567 

7541 

6698 

7426 

6826 

7308 

6953 

7187 

57 

4 

6437 

7653 

6569 

7539 

6700 

7424 

6828 

7306 

6955 

7185 

56 

5 

6439 

7651 

6572 

7538 

6702 

7422 

6831 

7304 

6957 

7183 

55 

6 

6441 

7649 

6574 

7536 

6704 

7420 

6833 

7302 

6959 

7181 

54 

7 

6443 

7647 

6576 

7534 

6706 

7418 

6835 

7300 

6961 

7179 

53 

8 

6446 

7645 

6578 

7532 

6709 

7416 

6837 

7298 

6963 

7177 

52 

9 

6448 

7644 

6580 

7530 

6711 

7414 

6839 

7296 

6965 

7175 

51 

io 

6450 

7642 

6583 

7528 

6713 

7412 

6841 

7294 

6967 

7173 

50 

11 

6452 

7640 

6585 

7526 

6715 

7410 

6843 

7292 

6970 

7171 

49 

12 

6455 

7638 

6587 

7524 

6717 

7408 

6845 

7290 

6972 

7169 

48 

13 

6457 

7636 

6589 

7522 

6719 

7406 

6848 

7288 

6974 

7167 

47 

14 

6459 

7634 

6591 

7520 

6722 

7404 

6850 

7286 

6976 

7165 

46 

15 

6461 

7632 

6593 

7518 

6724 

7402 

6852 

7284 

6978 

7163 

45 

16 

6463 

7630 

6596 

7516 

6726 

7400 

6854 

7282 

6980 

7161 

44 

17 

6466 

7629 

6598 

7515 

6728 

7398 

6856 

7280 

6982 

7159 

43 

18 

6468 

7627 

6600 

7513 

6730 

7396 

6S58 

7278 

6984 

7157 

42 

19 

6470 

7625 

6602 

7511 

6732 

7394 

6860 

7276 

6986 

7155 

41 

20 

6472 

7623 

6604 

7509 

6734 

7392 

6862 

7274 

6988 

7153 

40 

21 

6475 

7621 

6607 

7507 

6737 

7390 

6865 

7272 

6990 

7151 

39 

22 

6477 

7619 

6609 

7505 

6739 

7388 

6S67 

7270 

6992 

7149 

38 

23 

6479 

7617 

6611 

7503 

6741 

7387 

6869 

7268 

6995 

7147 

37 

24 

6481 

7615 

6613 

7501 

6743 

73S5 

6S71 

7266 

6997 

7145 

36 

25 

6483 

7613 

6615 

7499 

6745 

73S3 

6873 

7264 

6999 

7143 

35 

26 

6486 

7612 

6617 

7497 

6747 

7381 

6875 

7262 

7001 

7141 

34 

27 

6488 

7610 

6620 

7495 

6749 

7379 

6877 

7260 

7003 

7139 

33 

28 

6490 

7608 

6622 

7493 

6752 

7377 

6879 

7258 

7005 

7137 

32 

29 

6492 

7606 

6624 

7491 

6754 

7375 

6881 

7256 

7007 

7135 

31 

30 

6494 

7604 

6626 

7490 

6756 

7373 

6884 

7254 

7009 

7133 

30 

31 

6497 

7602 

6628 

7488 

6758 

7371 

6886 

7252 

7011 

7130 

29 

32 

6499 

7600 

6631 

7486 

6760 

7369 

6888 

7250 

7013 

7128 

28 

33 

6501 

7598 

6633 

7484 

6762 

7367 

6890 

7248 

7015 

7126 

27 

34 

6503 

7596 

6635 

7482 

6764 

7365 

6892 

7246 

7017 

7124 

26 

35 

6506 

7595 

6637 

7480 

6767 

7363 

6894 

7244 

7019 

7122 

25 

36 

6508 

7593 

6639 

7478 

6769 

7361 

6896 

7242 

7022 

7120 

24 

37 

6510 

7591 

6641 

7476 

6771 

7359 

6898 

7240 

7024 

7118 

23 

38 

'  6512 

7589 

6644 

7474 

6773 

7357 

6900 

7238 

7026 

7116 

22 

39 

6514 

7587 

6646 

7472 

6775 

7355 

6903 

7236 

7028 

7114 

21 

40 

6517 

75S5 

6648 

7470 

6777 

7253 

6905 

7234 

7030 

7112 

20 

41 

6519 

7583 

6650 

7468 

6779 

7351 

6907 

7232 

7032 

7110 

19 

42 

6521 

7581 

6652 

7466 

6782 

7349 

6909 

7230 

7034 

7108 

13 

43 

6523 

7579 

6654 

7464 

6784 

7347 

6911 

7228 

7036 

7106 

17 

44 

6525 

7578 

6657 

7463 

6786 

7345 

6913 

7226 

7038 

7104 

16 

45 

6528 

7576 

6659 

7461 

6788 

7343 

6915 

7224 

7040 

7102 

15 

46 

6530 

7574 

6661 

7459 

6790 

7341 

6917 

7222 

7042 

7100 

14 

47 

6532 

7572 

6663 

7457 

6792 

7339 

6919 

7220 

7044 

7098 

13 

48 

6534 

7570 

6665 

7455 

6794 

7337 

6921 

7218 

7046 

7096 

12 

49 

6536 

7568 

6667 

7453 

6797 

7335 

6924 

7216 

7048 

7094 

11 

50 

6539 

7566 

6670 

7451 

6799 

7333 

6926 

7214 

7050 

7092 

IO 

51 

6541 

7564 

6672 

7449 

6801 

7331 

6928 

7212 

7053 

7090 

9 

52 

6543 

7562 

6674 

7447 

6S03 

7329 

6930 

7210 

7055 

7088 

8 

53 

6545 

7560 

6676 

7445 

6805 

7327 

6932 

7208 

7057 

7085 

7 

54 

6547 

7559 

6678 

7443 

6807 

7325 

6934 

7206 

7059 

7083 

6 

55 

6550 

7557 

6680 

7441 

6809 

7323 

6936 

7203 

7061 

7081 

5 

56 

6552 

7555 

6683 

7439 

6811 

7321 

6938 

7201 

7063 

7079 

4 

57 

6554 

7553 

6685 

7437 

6814 

7319 

6940 

7199 

7065 

7077 

3 

58 

6556 

7551 

6687 

7435 

6816 

7318 

6942 

7197 

7067 

7075 

2 

59 

6558 

7549 

6689 

7433 

6818 

7316 

6944 

7195 

7069 

7073 

1 

60 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

7071 

7071 

O 

cos   sin 
49° 

cos  sin 
48° 

cos  sin 

47° 

cos  sin 
46° 

cos  sin 

45° 

t 

r 

NATURAL  TANGENTS  AND  COTANGENTS.  59 


t 

o°    - 

1° 

2° 

3° 

4° 

r 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

0000 

Infinite 

0175 

57.2900 

0349 

28.6363 

0524 

19.0811 

0699 

14.3007 

60 

1 

0003 

3437.75 

0177 

56.3506 

0352 

28.3994 

0527 

18.9755 

0702 

14.2411 

59 

2 

0006 

1718.87 

0180 

55.4415 

0355 

28.1664 

0530 

188711 

0705 

14.1821 

58 

3 

0009 

1145.92 

0183 

54.5613 

0358 

27.9372 

0533 

18.7678 

0708 

14.1235 

57 

4 

0012 

859.436 

0186 

53.7086 

0361 

27.7117 

0536 

18.6656 

0711 

14.0655 

56 

5 

0015 

687.549 

0189 

52.8821 

0364 

27.4899 

0539 

18.5645 

0714 

14.0079 

55 

6 

0017 

572.957 

0192 

52.0807 

0367 

27.2715 

0542 

18.4645 

0717 

13.9507 

54 

7 

0020 

491.106 

0195 

51.3032 

0370 

27.0566 

0544 

1S.3655 

0720 

13.S940 

53 

8 

0023 

429.718 

0198 

50.5485 

0373 

26.8450 

0547 

1S.2677 

0723 

13.8378 

52 

9 

0026 

381.971 

0201 

49.8157 

0375 

26.6367 

0550 

18.1708 

0726 

13.7821 

51 

io 

0029 

343.774 

0204 

49.1039 

0378 

26.4316 

0553 

18.0750 

0729 

13.7267 

50 

11 

0032 

312.521 

0207 

48.4121 

0381 

26.2296 

0556 

17.9S02 

0731 

13.6719 

49 

12 

0035 

286.478 

0209 

47.7395 

0384 

26.0307 

0559 

17.8863 

0734 

13.6174 

48 

13 

0038 

264.441 

0212 

47.0853 

0387 

25.8348 

0562 

17.7934 

0737 

13.5634 

47 

14 

0041 

245.552 

0215 

46.4489 

0390 

25.6418 

0565 

17.7015 

0740 

13.5098 

46 

15 

0044 

229. 1S2 

0218 

45.8294 

0393 

25.4517 

0568 

17.6106 

0743 

13.4566 

45 

16 

0047 

214.858 

0221 

45.2261 

0396 

25.2644 

0571 

17.5205 

0746 

13.4039 

44 

17 

0049 

202.219 

0224 

44.6386 

0399 

25.0798 

0574 

17.4314 

0749 

13.3515 

43 

18 

0052 

190.984 

0227 

44.0661 

0402 

24.8978 

0577 

17.3432 

0752 

13.2996 

42 

19 

0055 

180.932 

0230 

43.5081 

0405 

24.71S5 

0580 

17.2558 

0755 

13.2480 

41 

20 

0058 

171.885 

0233 

42.9641 

0407 

24.5418 

0582 

17.1693 

0758 

13.1969 

40 

21 

0061 

163.700 

0236 

42.4335 

0410 

24.3675 

0585 

17.0837 

0761 

13.1461 

39 

22 

0064 

156.259 

0239 

41.9158 

0413 

24.1957 

0588 

16.9990 

0764 

13.0958 

38 

23 

0067 

149.465 

0241 

41.4106 

0416 

24.0263 

0591 

16.9150 

0767 

13.0458 

37 

24 

0070 

143.237 

0244 

40.9174 

0419 

23.8593 

0594 

16.8319 

0769 

12.9962 

36 

25 

0073 

137.507 

0247 

40.4358 

0422 

23.6945 

0597 

16.7496 

0772 

12.9469 

35 

26 

0076 

132.219 

0250 

39.9655 

0425 

23.5321 

0600 

16.6681 

0775 

12.8981 

34 

27 

0079 

127.321 

0253 

39.5059 

0428 

23.3718 

0603 

16.5874 

0778 

12.8496 

33 

28 

0081 

122.774 

0256 

39.0568 

0431 

23.2137 

0606 

16.5075 

0781 

12.8014 

32 

29 

0084 

118.540 

0259 

38.6177 

0434 

23.0577 

0609 

16.42S3 

07S4 

12.7536 

31 

30 

0087 

114.589 

0262 

38.1885 

0437 

22.9038 

0612 

16  3499 

0787 

12.7062 

30 

31 

0090 

110.892 

0265 

37.7686 

0440 

22.7519 

0615 

16.2722 

0790 

12.6591 

29 

32 

0093 

107.426 

0268 

37.3579 

0442 

22.6020 

0617 

16.1952 

0793 

12.6124 

28 

33 

0096 

104.171 

0271 

36.9560 

0445 

224541 

0620 

16.1190 

0796 

12.5660 

27 

34 

0099 

101.107 

0274 

36.5627 

0448 

22.3081 

0623 

16.0435 

0799 

12.5199 

26 

35 

0102 

98.2179 

0276 

36.1776 

0451 

22.1640 

0626 

15.9687 

0S02 

12.4742 

25 

36 

0105 

95.4895 

0279 

35  8006 

0454 

22.0217 

0629 

15.8945 

0805 

12.4288 

24 

37 

0108 

92.9085 

0282 

35.4313 

0457 

21.8813 

0632 

15.8211 

0S08 

12.3838 

23 

38 

0111 

90.4633 

0285 

35.0695 

0460 

21.7426 

0635 

15.7483 

0810 

12.3390 

22 

39 

0113 

88.1436 

02S8 

34.7151 

0463 

21.6056 

0638 

15.6762 

0813 

12.2946 

21 

40 

0116 

85.9398 

0291 

34.3678 

.0466 

21.4704 

0641 

15.6048 

0816 

12.2505 

20 

41 

0119 

83.8435 

0294 

34.0273 

0469 

21.3369 

0644 

15.5340 

0819 

12.2067 

19 

42 

0122 

81.8470 

0297 

33.6935 

0472 

21.2049 

0647 

15.4638 

0822 

12.1632 

18 

43 

0125 

79.9434 

0300 

33.3662 

0475 

21.0747 

0650 

15.3943 

0825 

12.1201 

17 

44 

0128 

78.1263 

0303 

33.0452 

0477 

20.9460 

0653 

15.3254 

0828 

12.0772 

16 

45 

0131 

76.3900 

0306 

32.7305 

0480 

20.8188 

0655 

15.2571 

0831 

12.0346 

15 

46 

0134 

74.7292 

0308 

32.4213 

0483 

20.6932 

0658 

15.1893 

0834 

11.9923 

14 

47 

0137 

73.1390 

0311 

32.1181 

04S6 

20.5691 

0661 

15.1222 

0837 

11.9504 

13 

48 

0140 

71.6151 

0314 

31.8205 

0489 

20.4465 

0664 

15.0557 

0840 

11.9087 

12 

49 

0143 

70.1533 

0317 

31.5284 

0492 

20.3253 

0667 

14.9898 

0843 

11.8673 

11 

50 

0146 

68.7501 

0320 

31.2416 

0495 

20.2056 

0670 

14.9244 

0846 

11.8262 

IO 

51 

0148 

67.4019 

0323 

30.9599 

0498 

20.0872 

0673 

14.8596 

0849 

11.7853 

9 

52 

0151 

66.1055 

0326 

30.6833 

0501 

19.9702 

0676 

14.7954 

0851 

11.7448 

8 

53 

0154 

64.85S0 

0329 

30.4116 

0504 

19.8546 

0679 

14.7317 

OS54 

11.7045 

7 

54 

0157 

63.6567 

0332 

30.1446 

0507 

19.7403 

0682 

14.6685 

0857 

116645 

6 

55 

0160 

62.4992 

0335 

29.8823 

0509 

19.6273 

06S5 

14.6059 

0860 

11.6248 

5 

56 

0163 

61.3829 

0338 

29.6245 

0512 

19.5156 

0688 

14.5438 

0863 

11.5853 

4 

57 

0166 

60.3058 

0340 

29.3711 

0515 

19.4051 

0690 

14.4823 

0866 

11.5461 

3 

58 

0169 

59.2659 

0343 

29.1220 

0518 

19.2959 

0693 

14.4212 

0869 

11.5072 

2 

59 

0172 

58.2612 

0346 

28.8771 

0521 

19.1879 

0696 

14.3607 

0S72 

11.4685 

1 

60 

0175 

57.2900 

0349 

28.6363 

0524 

19.0S11 

0699 

14.3007 

0875 

11.4301 

O 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

r 

89° 

88° 

87° 

86° 

85° 

t 

60 

NATURAL  TANGENTS  AND 

COTANGENTS. 

t 

5° 

6° 

7° 

8° 

-  9° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan   cot 

tan 

cot 

O 

0875 

11.4301 

1051 

9.5144 

1228 

8.1443 

1405  7.1154 

1584 

6.3138 

60 

1 

0878 

11.3919 

1054 

9.4S78 

1231 

8.1248 

1408  7.1004 

1587 

6.3019 

59 

2 

0881 

11.3540 

1057 

9.4614 

1234 

8.1054 

1411  70855 

1590 

6.2901 

58 

3 

0884 

11.3163 

1060 

9.4352 

1237 

8.0860 

1414  7.0706 

1593 

6.2783 

57 

4 

0887 

11.2789 

1063 

9.4090 

1240 

8.0667 

1417  7.0558 

1596 

62666 

56 

5 

0890 

11.2417 

1066 

9.3831 

1243 

8.0476 

1420  7.0410 

1599 

6.2549 

55 

6 

0892 

11.2048 

1069 

9.3572 

1246 

8.0285 

1423  7.0264 

1602 

6.2432 

54 

7 

0895 

11.1681 

1072 

9.3315 

1249 

8.0095 

1426  7.0117 

1605 

6.2316 

53 

8 

0898 

11.1316 

1075 

9.3060 

1251 

7.9906 

1429  6.9972 

1608 

62200 

52 

9 

0901 

11.0954 

1078 

9.2806 

1254 

7.9718 

1432  6.9827 

1611 

6.2085 

51 

io 

0904 

11.0594 

1080 

9.2553 

1257 

7.9530 

1435  6.9682 

1614 

6.1970 

50 

11 

0907 

11.0237 

1083 

9.2302 

1260 

7.9344 

1438  6.9538 

1617 

6.1856 

49 

12 

0910 

10.9882 

1086 

9.2052 

1263 

7.9158 

1441  6.9395 

1620 

6.1742 

48 

13 

0913 

10.9529 

1089 

9.1803 

1266 

7.8973 

1444  6.9252 

1623 

6.1628 

47 

14 

0916 

10.9178 

1092 

9.1555 

1269 

7.8789 

1447  6.9110 

1626 

6.1515 

46 

15 

0919 

10.8829 

1095 

9.1309 

1272 

7.8606 

1450  6.8969 

1629 

6.1402 

45 

16 

0922 

10.8483 

1098 

9.1065 

1275 

7.8424 

1453  6.8828 

1632 

6.1290 

44 

17 

0925 

10.8139 

1101 

9.0821 

1278 

7.8243 

1456  6.8687 

1635 

6.1178 

43 

18 

0928 

10.7797 

1104 

9.0579 

1281 

7.8062 

1459  6.8548 

1638 

6.1066 

42 

19 

0931 

10.7457 

1107 

9.0338 

1284 

7.7883 

1462  6.8408 

1641 

6.0955 

41 

20 

0934 

10.7119 

1110 

9.0098 

12S7 

7.7704 

1465  6.8269 

1644 

6.0844 

40 

21 

0936 

10.6783 

1113 

8.9860 

1290 

7.7525 

1468  6.8131 

1647 

6.0734 

39 

22 

0939 

10.6450 

1116 

8.9623 

1293 

7.7348 

1471  6.7994 

1650 

6.0624 

38 

23 

0942 

10.6118 

1119 

8.9387 

1296 

7.7171 

1474  6.7856 

1653 

6.0514 

37 

24 

0945 

10.5789 

1122 

8.9152 

1299 

7.6996 

1477  6.7720 

1655 

6.0405 

36 

25 

0948 

10.5462 

1125 

8.8919 

1302 

7.6821 

1480  6.7584 

1658 

6.0296 

35 

26 

0951 

10.5136 

1128 

8.8686 

1305 

7.6647 

14S3  6.7448 

1661 

6.0188 

34 

27 

0954 

10.4813 

1131 

8.8455 

1308 

7.6473 

14S6  6.7313 

1664 

6.0080 

33 

28 

0957 

10.4491 

1134 

8.8225 

1311 

7.6301 

1489  6.7179 

1667 

5.9972 

32 

29 

0960 

10.4172 

1136 

8.7996 

1314 

7.6129 

1492  6.7045 

1670 

5. 9865 

31 

30 

0963 

10.3854 

1139 

8.7769 

1317 

7.5958 

1495  6.6912 

1673 

5.9758 

30 

31 

0966 

10.3538 

1142 

8.7542 

1319 

7.5787 

1497  6.6779 

1676 

5.9651 

29 

32 

0969 

10.3224 

1145 

8.7317 

1322 

7.5618 

1500  6.6646 

1679 

5.9545 

28 

33 

0972 

10.2913 

1148 

8.7093 

1325 

7.5449 

1503  6.6514 

1682 

5.9439 

27 

34 

0975 

10.2602 

1151 

8.6870 

1328 

7.5281 

1506  6.6383 

1685 

5.9333 

26 

35 

0978 

10.2294 

1154 

8.6648 

1331 

7.5113 

1509  6.6252 

1688 

5.9228 

25 

36 

0981 

10.1988 

1157 

8.6427 

1334 

7.4947 

1512  6.6122 

1691 

5.9124 

24 

37 

0983 

10.1683 

1160 

8.6208 

1337 

7.4781 

1515  6.5992 

1694 

5.9019 

23 

38 

0986 

10.1381 

1163 

8.5989 

1340 

7.4615 

1518  6.5863 

1697 

5.8915 

22 

39 

0989 

10.1080 

1166 

8.5772 

1343 

7.4451 

1521  6.5734 

1700 

5.8811 

21 

40 

0992 

10.0780 

1169 

8.5555 

1346 

7.4287 

1524  6.5606 

1703 

5.8708 

20 

41 

0995 

10.0483 

1172 

8.5340 

1349 

7.4124 

1527  6.5478 

1706 

5.8605 

19 

42 

0998 

10.0187 

1175 

8.5126 

1352 

7.3962 

1530  6.5350 

1709 

5.8502 

18 

43 

1001 

9.9893 

1178 

8.4913 

1355 

7.3800 

1533  6.5223 

1712 

5.8400 

17 

44 

1004 

9.9601 

1181 

8.4701 

1358 

7.3639 

1536  6.5097 

1715 

5.8298 

16 

45 

1007 

9.9310 

1184 

8.4490 

1361 

7.3479 

1539  6.4971 

1718 

5.8197 

15 

46 

1010 

9.9021 

1187 

8.4280 

1364 

7.3319 

1542  6.4846 

1721 

5.8095 

14 

47 

1013 

9.8734 

1189 

8.4071 

1367 

7.3160 

1545  6.4721 

1724 

5.7994 

13 

48 

1016 

9.8448 

1192 

8.3863 

1370 

7.3002 

1548  6.4596 

1727 

5.7894 

12 

49 

1019 

9.8164 

1195 

8.3656 

1373 

7.2844 

1551  6.4472 

1730 

5.7794 

11 

50 

1022 

9.7882 

1198 

8.3450 

1376 

7.2687 

1554  6.4348 

1733 

5.7694 

IO 

51 

1025 

9.7601 

1201 

8.3245 

1379 

7.2531 

1557  6.4225 

1736 

5.7594 

9 

52 

1028 

9.7322 

1204 

8.3041 

1382 

7.2375 

1560  6.4103 

1739 

5.7495 

8 

53 

1030 

9.7044 

1207 

8.2838 

1385 

7.2220 

1563  6.3980 

1742 

5.7396 

7 

54 

1033 

9.6768 

1210 

8.2636 

1388 

7.2066 

1566  6.3859 

1745 

5.7297 

6 

55 

1036 

9.6499 

1213 

8.2434 

1391 

7.1912 

1569  6.3737 

1748 

5.7199 

5 

56 

1039 

9.6220 

1216 

8.2234 

1394 

7.1759 

1572  6.3617 

1751 

5.7101 

4 

57 

1042 

9.5949 

1219 

8.2035 

1397 

7.1607 

1575  6.3496 

1754 

5.7004 

3 

58 

1045 

9.5679 

1222 

8.1837 

1399 

7.1455 

1578  6.3376 

1757 

5.6906 

2 

59 

1048 

9.5411 

1225 

8.1640 

1402 

7.1304 

1581  6.3257 

1760 

5.6809 

1 

60 

1051 

9.5144 

1228 

8.1443 

1405 

7.1154 

1584  6.3138 

1763 

5.6713 

O 

cot   tan 
84° 

cot   tan 
83° 

cot   tan 

82° 

cot   tan 
81° 

cot   tan 
80° 

t 

/ 

NATURAL   TANGENTS   AND   COTANGENTS.  61 


t 

io° 

11° 

12° 

13° 

14° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

1763 

5.6713 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

60 

1 

1766 

5.6617 

1947 

5.1366 

2129 

4.6979 

2312 

4.3257 

2496 

4.0058 

59 

2 

1769 

5.6521 

1950 

5.1286 

2132 

4.6912 

2315 

4.3200 

2499 

4.0009 

58 

3 

1772 

5.6425 

1953 

5.1207 

2135 

4.6S45 

2318 

4.3143 

2503 

3.9959 

57 

4 

1775 

5.6330 

1956 

5.1128 

2138 

4.6779 

2321 

4.3086 

2506 

3.9910 

56 

5 

1778 

5.6234 

1959 

5.1049 

2141 

4.6712 

2324 

4.3029 

2509 

3.9861 

55 

6 

1781 

5.6140 

1962 

5.0970 

2144 

4.6646 

2327 

4.2972 

2512 

3.9812 

54 

7 

1784 

5.6045 

1965 

5.0892 

2147 

4.6580 

2330 

4.2916 

2515 

3.9763 

53 

8 

1787 

5.5951 

1968 

5.0814 

2150 

4.6514 

2333 

4.2859 

2518 

39714 

52 

9 

1790 

5.5857 

1971 

5.0736 

2153 

4.6448 

2336 

4.2803 

2521 

3.9665 

51 

io 

1793 

5.5764 

1974 

5.0658 

2156 

4.6382 

2339 

4.2747 

2524 

3.9617 

50 

11 

1796 

5.5671 

1977 

5.0581 

2159 

4.6317 

2342 

4.2691 

2527 

3.9568 

49 

12 

1799 

5.5578 

1980 

5.0504 

2162 

4.6252 

2345 

4.2635 

2530 

3.9520 

48 

13 

1802 

5.5485 

1983 

50427 

2165 

4.6187 

2349 

4.2580 

2533 

3.9471 

47 

14 

1805 

5.5393 

1986 

5.0350 

2168 

4.6122 

2352 

4.2524 

2537 

3.9423 

46 

15 

1808 

5.5301 

1989 

5.0273 

2171 

4.6057 

2355 

4.2468 

2540 

3.9375 

45 

16 

1811 

5.5209 

1992 

5.0197 

2174 

4.5993 

2358 

4.2413 

2543 

3.9327 

44 

17 

1814 

5.5118 

1995 

5.0121 

2177 

4.5928 

2361 

4.2358 

2546 

3.9279 

43 

18 

1817 

5.5026 

1998 

5.0045 

2180 

4.5864 

2364 

4.2303 

2549 

3.9232 

42 

19 

1820 

5.4936 

2001 

4.9969 

2183 

4.5800 

2367 

4.2248 

2552 

3.9184 

41 

20 

1823 

5.4845 

2004 

4.9894 

2186 

4.5736 

2370 

4.2193 

2555 

3.9136 

40 

21 

1826 

5.4755 

2007 

4.9819 

2189 

4.5673 

2373 

4.2139 

2558 

3.9089 

39 

22 

1829 

5.4665 

2010 

4.9744 

2193 

4.5609 

2376 

4.2084 

2561 

3.9042 

38 

23 

1832 

5.4575 

2013 

4.9669 

2196 

4.5546 

2379 

4.2030 

2564 

3.8995 

37 

24 

1835 

5.4486 

2016 

4.9594 

2199 

4.5483 

2382 

4.1976 

2568 

3.8947 

36 

25 

1838 

5.4397 

2019 

4.9520 

2202 

4.5420 

2385 

4.1922 

2571 

3.8900 

35 

26 

1841 

5.4308 

2022 

4.9446 

2205 

4.5357 

2388 

4.1868 

2574 

3.8854 

34 

27 

1844 

5.4219 

2025 

4.9372 

2208 

4.5294 

2392 

4.1814 

2577 

3.8807 

33 

28 

1847 

5.4131 

2028 

4.9298 

2211 

4.5232 

2395 

4.1760 

2580 

3.8760 

32 

29 

1850 

5.4043 

2031 

4.9225 

2214 

4.5169 

2398 

4.1706 

2583 

3.8714 

31 

30 

1853 

5.3955 

2035 

4.9152 

2217 

4.5107 

2401 

4.1653 

2586 

3.8667 

30 

31 

1856 

5.3868 

2038 

4.9078 

2220 

4.5045 

2404 

4.1600 

2589 

3.8621 

29 

32 

1859 

5.3781 

2941 

4.9006 

2223 

4.4983 

2407 

4.1547 

2592 

3.8575 

28 

33 

1862 

5.3694 

2044 

4.8933 

2226 

4.4922 

2410 

4.1493 

2595 

3.8528 

27 

34 

1865 

5.3607 

2047 

4.8860 

2229 

4.4860 

2413 

4.1441 

2599 

3.8482 

26 

35 

1868 

5.3521 

2050 

4.8788 

2232 

4.4799 

2416 

4.1388 

2602 

3.8436 

25 

36 

1871 

5.3435 

2053 

4.8716 

2235 

4.4737 

2419 

4.1335 

2605 

3.S391 

24 

37 

1874 

5.3349 

2056 

4.8644 

2238 

4.4676 

2422 

4.1282 

2608 

3.8345 

23 

38 

1877 

5.3263 

2059 

4.8573 

2241 

4.4615 

2425 

4.1230 

2611 

3.8299 

22 

39 

1880 

5.3178 

2062 

4.8501 

2244 

4.4555 

2428 

4.1178 

2614 

3.8254 

21 

40 

1883 

5.3093 

2065 

4.8430 

2247 

4.4494 

2432 

4.1126 

2617 

3.8208 

20 

41 

1887 

5.3008 

2068 

4.8359 

2251 

4.4434 

2435 

4.1074 

2620 

3.8163 

19 

42 

1890 

5.2924 

2071 

4.8288 

2254 

4.4374 

2438 

4.1022 

2623 

3.8118 

18 

43 

1893 

5.2839 

2074 

4.8218 

2257 

4.4313 

2441 

4.0970 

2627 

3.8073 

17 

44 

1896 

5.2755 

2077 

4.8147 

2260 

4.4253 

2444 

4.0918 

2630 

3.8028 

16 

45 

1899 

5.2672 

2080 

4.8077 

2263 

4.4194 

2447 

4.0867 

2633 

3.7983 

15 

46 

1902 

5.2588 

2083 

4.8007 

2266 

4.4134 

2450 

4.0815 

2636 

3.7938 

14 

47 

1905 

5.2505 

2086 

4.7937 

2269 

4.4075 

2453 

4.0764 

2639 

3.7893 

13 

48 

1908 

5.2422 

2089 

4.7867 

2272 

4.4015 

2456 

4.0713- 

2642 

3.7848 

12 

49 

1911 

5.2339 

2092 

4.7798 

2275 

4.3956 

2459 

4.0662 

2645 

3.7804 

11 

50 

1914 

5.2257 

2095 

4.7729 

2278 

4.3897 

2462 

4.0611 

2648 

3.7760 

IO 

51 

1917 

5.2174 

2098 

4.7659 

2281 

4.3838 

2465 

4.0560 

2651 

3.7715 

9 

52 

1920 

5.2092 

2101 

4.7591 

2284 

4.3779 

2469 

4.0509 

2655 

3.7671 

8 

53 

1923 

5.2011 

2104 

4.7522 

2287 

4.3721 

2472 

4.0459 

2658 

3.7627 

7 

54 

1926 

5.1929 

2107 

4.7453 

2290 

4.3662 

2475 

4.0408 

2661 

3.7583 

6 

55 

1929 

5.1848 

2110 

4.7385 

2293 

4.3604 

2478 

4.0358 

2664 

3.7539 

5 

56 

1932 

5.1767 

2113 

4.7317 

2296 

4.3546 

2481 

4.0308 

2667 

3.7495 

4 

57 

1935 

5.1686 

2116 

4.7249 

2299 

4.3488 

2484 

4.0257 

2670 

3.7451 

3 

58 

1938 

5.1606 

2119 

4.7181 

2303 

4.3430 

2487 

4.0207 

2673 

3.7408 

2 

59 

1941 

5.1526 

2123 

4.7114 

2306 

4.3372 

2490 

4.0158 

2676 

3.7364 

1 

GO 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

2679 

3.7321 

O 

cot   tan 
79° 

cot   tan 

78° 

cot   tan 

77° 

cot   tan  • 
76° 

cot   tan 

75° 

t 

t 

62 


NATURAL   TANGENTS   AND   COTANGENTS. 


r 

15° 

16° 

17° 

18° 

19° 

/ 

tan 

cot 

tan 

cot 

tan   cot 

tan 

cot 

tan 

cot 

O 

2679 

3.7321 

2867 

3.4874 

3057  3.2709 

3249 

3.0777 

3443 

2.9042 

60 

1 

2683 

3.7277 

2871 

3.4836 

3060  3.2675 

3252 

3.0746 

3447 

2.9015 

59 

2 

2686 

3.7234 

2874 

3.4798 

3064  3.2641 

3256 

3.0716 

3450 

2.8987 

58 

3 

2689 

3.7191 

'2877 

3.4760 

3067  3.2607 

3259 

3.0686 

3453 

2.8960 

57 

4 

2692 

3.7148 

2880 

3.4722 

3070  3.2573 

3262 

3.0655 

3456 

2.8933 

56 

5 

2695 

3.7105 

2883 

3.4684 

3073  3.2539 

3265 

3.0625 

3460 

2.8905 

55 

6 

2698 

3.7062 

2886 

3.4646 

3076  3.2506 

3269 

3.0595 

3463 

2.8878 

54 

7 

2701 

3.7019 

2890 

3.4608 

3080  3.2472 

3272 

3.0565 

3466 

2.8851 

53 

8 

2704 

3.6976 

2893 

3.4570 

3083  3.2438 

3275 

3.0535 

3469 

2.8824 

52 

9 

2708 

3.6933 

2896 

3.4533 

3086  3.2405 

3278 

3.0505 

3473 

2.8797 

51 

io 

2711 

3.6891 

2899 

3.4495 

3089  3.2371 

3281 

3.0475 

3476 

2.8770 

50 

11 

2714 

3.6848 

2902 

3.4458 

3092  3.2338 

3285 

3.0445 

3479 

2.8743 

49 

12 

2717 

3.6806 

2905 

3.4420 

3096  3.2305 

3288 

3.0415 

3482 

2.8716 

48 

13 

2720 

3.6764 

2908 

3.4383 

3099  3.2272 

3291 

3.0385 

3486 

2.8689 

47 

14 

2723 

3.6722 

2912 

3.4346 

3102  3.2238 

3294 

3.0356 

3489 

2.8662 

46 

15 

2726 

3.6680 

2915 

3.4308 

3105  3.2205 

3298 

3.0326 

3492 

2.8636 

45 

16 

2729 

3.6638 

2918 

3.4271 

3108  3.2172 

3301 

3.0296 

3495 

2.8609 

44 

17 

2733 

3.6596 

2921 

3.4234 

3111  3.2139 

3304 

3.0267 

3499 

2.8582 

43 

18 

2736 

3.6554 

2924 

3.4197 

3115  3.2106 

3307 

3.0237 

3502 

2.8556 

42 

19 

2739 

3.6512 

2927 

3.4160 

3118  3.2073 

3310 

3.0208 

3505 

2.8529 

41 

20 

2742 

3.6470 

2931 

3.4124 

3121  3.2041 

3314 

3.0178 

3508 

2.8502 

40 

21 

2745 

3.6429 

2934 

3.4087 

3124  3.2008 

3317 

3.0149 

3512 

2.8476 

39 

22 

2748 

3.6387 

2937 

3.4050 

3127  3.1975 

3320 

3.0120 

3515 

2.8449 

38 

23 

2751 

3.6346 

2940 

3.4014 

3131  3.1943 

3323 

3.0090 

3518 

2.8423 

37 

24 

2754 

3.6305 

2943 

3.3977 

3134  3.1910 

3327 

3.0061 

3522 

2.8397 

36 

25 

275$  3.6264 

2946 

3.3941 

3137  3.1878 

3330 

3.0032 

3525 

2.8370 

35 

26 

2761 

3  6222 

2949 

3.3904 

3140  3.1845 

3333>  3.0003 

3528 

2.8344 

34 

27 

2764 

3.6181 

2953 

3.3868 

3143  3.1813 

3336 

2.9974 

3531 

2.8318 

33 

28 

2767 

3.6140 

2956 

3.3832 

3147  3.1780 

3339 

2.9945 

3535 

2.8291 

32 

29 

2770 

3.6100 

2959 

3.3796 

3150  3.1748 

3343 

2.9916 

3538 

2.8265 

31 

30 

2773 

3.6059 

2962 

3.3759 

3153  3.1716 

3346 

2.9887 

3541 

2.8239 

30 

31 

2776 

3.6018 

2965 

3.3723 

3156  3.1684 

3349 

2.9858 

3544 

2.8213 

29 

32 

2780 

3.5978 

2968 

3.3687 

3159  3.1652 

3352 

2.9829 

3548 

2.8187 

28 

33 

2783 

3.5937 

2972 

3.3652 

3163  3.1620 

3356 

2.9800 

3551 

2.8161 

27 

34 

2786 

3.5897 

2975 

3.3616 

3166  3.1588 

3359 

2.9772 

3554 

2.8135 

26 

35 

2789 

3.5856 

2978 

3.3580 

3169  3.1556 

3362 

2.9743 

3558 

2.8109 

25 

36 

2792 

3.5816 

2981 

3.3544 

3172  3.1524 

3365 

2.9714 

3561 

2.8083 

24 

37 

2795 

3.5776 

2984 

3.3509 

3175  3.1492 

3369 

2.9686 

3564 

2.8057 

23 

38 

2798 

3.5736 

2987 

3.3473 

3179  3.1460 

3372 

2.9657 

3567 

2.8032 

22 

39 

2801 

3.5696 

2991 

3.3438 

3182  3.1429 

3375 

2.9629 

3571 

2.8006 

21 

40 

2805 

3.5656 

2994 

3.3402 

3185  3.1397 

3378 

2.9600 

3574 

2.7980 

20 

41 

2808 

3.5616 

2997 

3.3367 

3188  3.1366 

3382 

2.9572 

3577 

2.7955 

19 

42 

2811 

3.5576 

3000 

3.3332 

3191  3.1334 

3385 

2.9544 

3581 

2.7929  • 

18 

43 

2814 

3.5536 

3003 

3.3297 

3195  3.1303 

3388 

2.9515 

3584 

2.7903 

17 

44 

2817 

3.5497 

3006 

3.3261 

3198  3.1271 

3391 

2.9487 

3587 

2.7878 

16 

45 

2820 

3.5457 

3010 

3.3226 

3201  3.1240 

3395 

2.9459 

3590 

2.7852 

15 

46 

2823 

3.5418 

3013 

3.3191 

3204  3.1209 

3398 

2.9431 

3594 

2.7827 

14 

47 

2827 

3.5379 

3016 

3.3156 

3207  3.1178 

3401 

2.9403 

3597 

2.7801 

13 

48 

2830 

3.5339 

3019 

3.3122 

3211  3.1146 

3404 

2.9375 

3600 

2.7776 

12 

49 

2833 

3.5300 

3022 

3.3087 

3214  3.1115 

3408 

2.9347 

3604 

2.7751 

11 

50 

2836 

3.5261 

3026 

3.3052 

3217  3.1084 

3411 

2.9319 

3607 

2.7725 

IO 

51 

2839 

3.5222 

3029 

3.3017 

3220  3.1053 

3414 

2.9291 

3610 

2.7700 

9 

52 

2842 

3.5183 

3032 

3.2983 

3223  3.1022 

3417 

2.9263 

3613 

2.7675 

8 

53 

2845 

3.5144 

3035 

3.2948 

3227  3.0991 

3421 

2.9235 

3617 

2.7650 

7 

54 

2849 

3.5105 

3038 

3.2914 

3230  3.0961 

3424 

2.9208 

3620 

2.7625 

6 

55 

2852 

3.5067 

3041 

3.28S0 

3233  3.0930 

3427 

2.9180 

3623 

2.7600 

5 

56 

2855 

3  5028 

3045 

3.2845 

3236  3.0899 

3430 

2.9152 

3627 

2.7575 

4 

57 

2858 

3.4989 

3048 

3.2811 

3240  3.0868 

3434 

2.9125 

3630 

2.7550 

3 

58 

2861 

3.4951 

3051 

3.2777 

3243  3.0838 

3437 

2.9097 

3633 

2.7525 

2 

59 

2864 

3.4912 

3054 

3.2743 

3246  3.0807 

3440 

2.9070 

3636 

2.7500 

1 

60 

2867 

3.4874 

3057 

3.2709 

3249  3.0777 

3443 

2.9042 

3640 

2.7475 

O 

cot 

tan 

cot 

tan 

cot   tan 

cot 

tan 

cot 

tan 

f 

74° 

73° 

72° 

71° 

70° 

/ 

NATURAL  TANGENTS  AND 

COTANGENTS. 

t>3 

r 

20° 

21° 

22° 

23° 

24° 

/ 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

3640 

2.7475 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

60 

1 

3643 

2.7450 

3842 

2.6028 

4044 

2.4730 

4248 

2.3539 

4456 

2.2443 

59 

2 

3646 

2.7425 

3845 

2.6006 

4047 

2.4709 

4252 

2.3520 

4459 

2.2425 

58 

3 

3650 

2.7400 

3849 

2.5983 

4050 

2.4689 

4255 

2.3501 

4463 

2.2408 

57 

4 

3653 

2.7376 

3852 

2.5961 

4054 

2.4668 

4258 

2.3483 

4466 

2.2390 

56 

5 

3656 

2.7351 

3855 

2.5938 

4057 

2.4648 

4262 

2.3464 

4470 

2.2373 

55 

6 

3659 

2.7326 

3859 

2.5916 

4061 

2.4627 

4265 

2.3445 

4473 

2.2355 

54 

7 

3663 

2.7302 

3862 

2.5893 

4064 

2.4606 

4269 

2.3426 

4477 

2.2338 

53 

8 

3666 

2.7277 

3865 

2.5871 

4067 

2.4586 

4272 

2.3407 

4480 

2.2320 

52 

9 

3669 

2.7253 

3869 

2.5848 

4071 

2.4566 

4276 

2.3388 

4484 

2.2303 

51 

io 

3673 

2.7228 

3872 

2.5826 

4074 

2.4545 

4279 

2.3369 

4487 

2.2286 

50 

11 

3676 

2.7204 

3875 

2.5804 

4078 

2.4525 

4283 

2.3351 

4491 

2.2268 

49 

12 

3679 

2.7179 

3879 

2.5782 

4081 

2.4504 

4286 

2.3332 

4494 

2.2251 

48 

13 

3683 

2.7155 

3882 

2.5759 

4084 

2.4484 

4289 

2.3313 

4498 

2.2234 

47 

14 

3686 

2.7130 

3885 

2.5737 

4088 

2.4464 

4293 

2.3294 

4501 

2.2216 

46 

15 

3689 

2.7106 

3889 

2.5715 

4091 

2.4443 

4296 

2.3276 

4505 

2.2199 

45 

16 

3693 

2.7082 

3892 

2.5693 

4095 

2.4423 

4300 

2.3257 

4508 

2.2182 

44 

17 

3696 

2.7058 

3895 

2.5671 

4098 

2.4403 

4303 

2.3238 

4512 

2.2165 

43 

18 

3699 

2.7034 

3899 

2.5649 

4101 

2.4383 

4307 

2.3220 

4515 

2.2148 

42 

19 

3702 

2.7009 

3902 

2.5627 

4105 

2.4362 

4310 

2.3201 

4519 

2.2130 

41 

20 

3706 

2.6985 

3906 

2.5605 

4108 

2.4342 

4314 

2.3183 

4522 

2.2113 

40 

21 

3709 

2.6961 

3909 

2.5533 

4111 

2.4322 

4317 

2.3164 

4526 

2.2096 

39 

22 

3712 

2.6937 

3912 

2.5561 

4115 

2.4302 

4320 

2.3146 

4529 

2.2079 

38 

23 

3716 

2.6913 

3916 

2.5539 

4118 

2.4282 

4324 

2.3127 

4533 

2.2062 

37 

24 

3719 

2.6889 

3919 

2.5517 

4122 

2.4262 

4327 

2.3109 

4536 

2.2045 

36 

25 

3722 

2.6865 

3922 

2.5495 

4125 

2.4242 

4331 

2.3090 

4540 

2.2028 

35 

26 

3726 

2.6841 

3926 

2.5473 

4129 

2.4222 

4334 

2.3072 

4543 

2.2011 

34 

27 

3729 

2.6818 

3929 

2.5452 

4132 

2.4202 

4338 

2.3053 

4547 

2.1994 

33 

28 

3732 

2.6794 

3932 

2.5430 

4135 

2.4182 

4341 

2.3035 

4550 

2.1977 

32 

29 

3736 

2.6770 

3936 

2.5408 

4139 

2.4162 

4345 

2.3017 

4554 

2.1960 

31 

30 

3739 

2.6746 

3939 

2.5386 

4142 

2.4142 

4348 

2.2998 

4557 

2.1943 

30 

31 

3742 

2.6723 

3942 

2.5365 

4146 

2.4122 

4352 

2.2980 

4561 

2.1926 

29 

32 

3745 

2.6699 

3946 

2.5343 

4149 

2.4102 

4355 

2.2962 

4564 

2.1909 

28 

33 

3749 

2.6675 

3949 

2.5322 

4152 

2.4083 

4359 

2.2944 

4568 

2.1892 

27 

34 

3752 

2.6652 

3953 

2.5300 

4156 

2.4063 

4362 

2.2925 

4571 

2.1876 

26 

35 

3755 

2.6628 

3956 

2.5279 

4159 

2.4043 

4365 

2.2907 

4575 

2.1859 

25 

36 

3759 

2.6605 

3959 

2.5257 

4163 

2.4023 

4369 

2.2889 

4578 

2.1842 

24 

37 

3762 

2.6581 

3963 

2.5236 

4166 

2.4004 

4372 

2.2871 

4582 

2.1825 

23 

38 

3765 

2.6558 

3966 

2.5214 

4169 

2.3984 

4376 

2.2853 

4585 

2.1808 

22 

39 

3769 

2.6534 

3969 

2.5193 

4173 

2.3964 

4379 

2.2835 

4589 

2.1792 

21 

40 

3772 

2.6511 

3973 

2.5172 

4176 

2.3945 

4383 

2.2817 

4592 

2.1775 

20 

41 

3775 

2.6488 

3976 

2.5150 

4180 

2.3925 

4386 

2.2799 

4596 

2.1758 

19 

42 

3779 

2.6464 

3979 

2.5129 

4183 

2.3906 

4390 

2.2781 

4599 

2.1742 

18 

43 

3782 

2.6441 

3983 

2.5108 

4187 

2.3886 

4393 

2.2763 

4603 

2.1725 

17 

44 

3785 

2.6418 

3986 

2.5086 

4190 

2.3867 

4397 

2.2745 

4607 

2.1708 

16 

45 

3789 

2.6395 

3990 

2.5065 

4193 

2.3847 

4400 

2.2727 

4610 

2.1692 

15 

46 

3792 

2.6371 

3993 

2.5044 

4197 

2.3828 

4404 

2.2709 

4614 

2.1675 

14 

47 

3795 

2.6348 

3996 

2.5023 

4200 

2.3808 

4407 

2.2691 

4617 

2.1659 

13 

48 

3799 

2.6325 

4000 

2.5002 

4204 

2.3789 

4411 

2.2673 

4621 

2.1642 

12 

49 

3802 

2.6302 

4003 

2.4981 

4207 

2.3770 

4414 

2.2655 

4624 

2.1625 

11 

50 

3805 

2.6279 

4006 

2.4960 

4210 

2.3750 

4417 

2.2637 

4628 

2.1609 

IO 

51 

3809 

2.6256 

4010 

2.4939 

4214 

2.3731 

4421 

2.2620 

4631 

2.1592 

9 

52 

3812 

2.6233 

4013 

2.4918 

4217 

2.3712 

4424 

2.2602 

4635 

2.1576 

8 

53 

3815 

2.6210 

4017 

2.4897 

4221 

2.3693 

4428 

2.2584 

4638 

2.1560 

7 

54 

3819 

2.6187 

4020 

2.4876 

4224 

2.3673 

4431 

2.2566 

4642 

2.1543 

6 

55 

3822 

2.6165 

4023 

2.4855 

4228 

2.3654 

4435 

2.2549 

4645 

2.1527 

5 

56 

3825 

2.6142 

4027 

2.4834 

4231 

2.3635 

4438 

2.2531 

4649 

2.1510 

4 

57 

3829 

2.6119 

4030 

2.4813 

4234 

2.3616 

4442 

2.2513 

4652 

2.1494 

3 

58 

3832 

2.6096 

4033 

2.4792 

4238 

2.3597 

4445 

2.2496 

4656 

2.1478 

2 

59 

3835 

2.6074 

4037 

2.4772 

4241 

2.3578 

4449 

2.2478 

4660 

2.1461 

1 

60 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

4663 

2.1445 

O 

cot   tan 
69° 

cot   tan 
68° 

cot   tan 
67° 

cot   tan 
66° 

cot   tan 
65° 

t 

t 

64 


NATURAL   TANGENTS   AND   COTANGENTS. 


t 

25° 

26° 

27° 

28° 

29° 

t 

tan   cot 

tan   cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

4663  2.1445 

4877  2.0503 

5095 

1.9626 

5317 

1.8807 

5543 

1.8040 

60 

1 

4667  2.1429 

4881  2.0488 

5099 

1.9612 

5321 

1.8794 

5547 

1.8028 

59 

2 

4670  2.1413 

4885  2.0473 

5103 

1.9598 

5325 

1.8781 

5551 

1.8016 

58 

3 

4674  2.1396 

4888  2.0458 

5106 

1.9584 

5328 

1.8768 

5555 

1.8003 

57 

4 

4677  2.1380 

4892  2.0443 

5110 

1.9570 

5332 

1.8755 

5558 

1.7991 

56 

5 

4681  2.1364 

4895  2.0428 

5114 

1.9556 

5336 

1.8741 

5562 

1.7979 

55 

6 

4684  2.1348 

4899  2.0413 

5117 

1.9542 

5340 

1.8728 

5566 

1.7966 

54 

7 

4688  2.1332 

4903  2.0398 

5121 

1.9528 

5343 

1.8715 

5570 

1.7954 

53 

8 

4691  2.1315 

4906  2.0383 

5125 

1.9514 

5347 

1.8702 

5574 

1.7942 

52 

9 

4695  2.1299 

4910  2.0368 

5128 

1.9500 

5351 

1.8689 

5577 

1.7930 

51 

io 

4699  2.1283 

4913  2.0353 

5132 

1.9486 

5354 

1.8676 

5581 

1.7917 

50 

11 

4702  2.1267 

4917  2.0338 

5136 

1.9472 

5358 

1.8663 

5585 

1.7905 

49 

12 

4706  2.1251 

4921  2.0323 

5139 

1.9458 

5362 

1.8650 

5589 

1.7893 

48 

13 

4709  2.1235 

4924  2.0308 

5143 

1.9444 

5366 

1.8637 

5593 

1.7881 

47 

14 

4713  2.1219 

4928  2.0293 

5147 

1.9430 

5369 

1.8624 

5596 

1.7868 

46 

15 

4716  2.1203 

4931  2.0278 

5150 

1.9416 

5373 

1.8611 

5600 

1.7856 

45 

16 

4720  2.1187 

4935  2.0263 

5154 

1.9402 

5377 

1.8598 

5604 

1.7844 

44 

17 

4723  2.1171 

4939  2.0248 

5158 

1.9388 

53S1 

1.8585 

5608 

1.7832 

43 

18 

4727  2.1155 

4942  2.0233 

5161 

1.9375 

5384 

1.8572 

5612 

1.7820 

42 

19 

4731  2.1139 

4946  2.0219 

5165 

1.9361 

5388 

1.8559 

5616 

1.7808 

41 

20 

4734  2.1123 

4950  2.0204 

5169 

1.9347 

5392 

1.8546 

5619 

1.7796 

40 

21 

4738  2.1107 

4953  2.0189 

5172 

1.9333 

5396 

1.8533 

5623 

1.7783 

39 

22 

4741  2.1092 

4957  2:0174 

5176 

1.9319 

5399 

1.8520 

5627 

1.7771 

38 

23 

4745  2.1076 

4960  2.0160 

5180 

1.9306 

5403 

1.8507 

5631 

1.7759 

37 

24 

4748  2.1060 

4964  .2.0145 

5184 

1.9292 

5407 

1.8495 

5635 

1.7747 

36 

25 

4752  2.1044 

4968  2.0130 

5187 

1.9278 

5411 

1.8482 

5639 

1.7735 

35 

26 

4755  2.1028 

4971  2.0115 

5191 

1.9265 

5415 

1.8469 

5642 

1.7723 

34 

27 

4759  2.1013 

4975  2.0101 

5195 

1.9251 

5418 

1.8456 

5646 

1.7711 

33 

28 

4763  2.0997 

4979  2.0086 

5198 

1.9237 

5422 

1.8443 

5650 

1.7699 

32 

29 

4766  2.0981 

4982  2.0072 

5202 

1.9223 

5426 

1.8430 

5654 

1.7687 

31 

30 

4770' 2.0965 

4986  2.0057 

5206 

1.9210 

5430 

1.8418 

5658 

1.7675 

30 

31 

4773  2.0950 

49S9  2.0042 

5209 

1.9196 

5433 

1.8405 

5662 

1.7663 

29 

32 

4777  2.0934 

4993  2.0028 

5213 

1.9183 

5437 

1.8392 

5665 

1.76S1 

28 

33 

4780  2.0918 

4997  2.0013 

5217 

1.9169 

5441 

1.8379 

5669 

1.7639 

27 

34 

4784  2.0903 

5000  1.9999 

5220 

1.9155 

5445 

1.8367 

5673 

1.7627 

26 

35 

4788  2.0887 

5004  1.9984 

5224 

1.9142 

5448 

1.8354 

5677 

1.7615 

25 

36 

4791  2.0872 

5008  1.9970 

5228 

1.9128 

5452 

1.8341 

5681 

1.7603 

24 

37 

4795  2.0856 

5011  1.9955 

5232 

1.9115 

5456 

1.8329 

5685 

1.7591 

23 

38 

4798  2.0S40 

5015  1.9941 

5235 

1.9101 

5460 

1.8316 

5688 

1.7579 

22 

39 

4802  2.0825 

5019  1.9926 

5239 

1.9088 

5464 

1.8303 

5692 

1.7567 

21 

40 

4806  2.0809 

5022  1.9912 

5243 

1.9074 

5467 

1.8291 

5696 

1.7556 

20 

41 

4809  2.0794 

5026  1.9897 

5246 

1.9061 

5471 

1.8278 

5700 

1.7544 

19 

42 

4813  2.0778 

5029  1.9883 

5250 

1.9047 

5475 

1.8265 

5704 

1.7532 

18 

43 

4816  2.0763 

5033  1.9868 

5254 

1.9034 

5479 

1.8253 

5708 

1.7520 

17 

44 

4820  2.0748 

5037  1.9854 

5258 

1.9020 

5482 

1.8240 

5712 

1.7508 

16 

45 

4823  2.0732 

5040  1.9840 

5261 

1.9007 

5486 

1.8228 

5715 

1.7496 

15 

46 

4827  2.0717 

5044  1.9825 

5265 

1.8993 

5490 

1.8215 

5719 

1.7485 

14 

47 

4831  2.0701 

5048  1.9811 

5269 

1.S980 

5494 

1.8202 

5723 

1.7473 

13 

48 

4834  2.0686 

5051  1.9797 

5272 

1.8967 

5498 

1.8190 

5727 

1.7461 

12 

49 

4838  2.0671 

5055  1.9782 

5276 

1.8953 

5501 

1.8177 

5731 

1.7449 

11 

50 

4841  2.0655 

5059  1.9768 

5280 

1.8940 

5505 

1.8165 

5735 

1.7437 

10 

51 

4845  2.0640 

5062  1.9754 

5284 

1.8927 

5509 

1.8152 

5739 

1.7426 

9 

52 

4849  2.0625 

5066  1.9740 

5287 

1.8913 

5513 

1.8140 

5743 

1.7414 

8 

53 

4852  2.0609 

5070  1.9725 

5291 

1.8900 

5517 

1.8127 

5746 

1.7402 

7 

54 

4856  2.0594 

5073  1.9711 

5295 

1.8887 

5520 

1.8115 

5750 

1.7391 

6 

55 

4859  2.0579 

5077  1.9697 

5298 

1.8873 

5524 

1.8103 

5754 

1.7379 

5 

56 

4863  2.0564 

5081  1.9683 

5302 

1.8860 

5528 

1.8090 

5758 

1.7367 

4 

57 

4867  2.0549 

5084  1.9669 

5306 

1.8847 

5532 

1.8078 

5762 

1.7355 

3 

58 

4870  2.0533 

5088  1.9654 

5310 

1.8834 

5535 

1.8065 

5766 

1.7344 

2 

59 

4874  2.0518 

5092  1.9640 

5313 

1.8820 

5539 

1.8053 

5770 

1.7332 

1 

60 

4877  2.0503 

5095  1.9626 

5317 

1.8807 

5543 

1.8040 

5774 

1.7321 

O 

r 

cot   tan 
64° 

cot   tan 
63° 

cot 

tan 

cot   tan 
61° 

cot  tan 
60° 

t 

62° 

NATURAL  TANGENTS  AND 

COTANGENTS. 

bb 

t 

30° 

31° 

32° 

33° 

34° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan   cot 

o 

5774 

1,7321 

6009 

1.6643 

6249 

1.6003 

6494 

1.5399 

6745  1.4826 

60 

1 

5777 

17309 

6013 

1.6632 

6253 

1.5993 

6498 

1.5389 

6749  1.4816 

59 

2 

5781 

1.7297 

6017 

1.6621 

6257 

1.5983 

6502 

1.5379 

6754  1.4807 

58 

3 

5785 

1.7286 

6020 

1.6610 

6261 

1.5972 

6506 

1.5369 

6758  1.4798 

57 

4 

5789 

1.7274 

6024 

1.6599 

6265 

1.5962 

6511 

1.5359 

6762  1.4788 

56 

5 

5793 

1.7262 

6028 

1.6588 

6269 

1.5952 

6515 

1.5350 

6766  1.4779 

55 

6 

5797 

1.7251 

6032 

1.6577 

6273 

1.5941 

6519 

1.5340 

6771  1.4770 

54 

7 

5801 

1.7239 

6036 

1.6566 

6277 

1.5931 

6523 

1.5330 

6775  1.4761 

53 

8 

5805 

1.7228 

6040 

1.6555 

6281 

1.5921 

6527 

1.5320 

6779  14751 

52 

9 

5808 

1.7216 

6044 

1.6545 

6285 

1.5911 

6531 

1.5311 

6783  1.4742 

51 

io 

5812 

1.7205 

6048 

1.6534 

6289 

1.5900 

6536 

1.5301 

6787  1.4733 

50 

11 

5816 

1.7193 

6052 

1.6523 

6293 

1.5890 

6540 

1.5291 

6792  1.4724 

49 

12 

5820 

1.7182 

6056 

1.6512 

6297 

1.5880 

6544 

1.5282 

6796  1.4715 

48 

13 

5824 

1.7170 

6060 

1.6501 

6301 

1.5869 

6548 

1.5272 

6800  1.4705 

47 

14 

5828 

1.7159 

6064 

1.6490 

6305 

1.5859 

6552 

1.5262 

6805  1.4696 

46 

15 

5832 

1.7147 

6068 

1.6479 

6310 

1.5849 

6556 

1.5253 

6809  1.4687 

45 

16 

5836 

1.7136 

6072 

1.6469 

6314 

1.5839 

6560 

1.5243 

6813  1.4678 

44 

17 

5840 

1.7124 

6076 

1.6458 

6318 

1.5829 

6565 

1.5233 

6817  1.4669 

43 

18 

5844 

1.7113 

6080 

1.6447 

6322 

1.5818 

6569 

1.5224 

6822  1.4659 

42 

19 

5847 

1.7102 

6084 

1.6436 

6326 

1.5808 

6573 

1.5214 

6826  1.4650 

41 

20 

5851 

1.7090 

6088 

1.6426 

6330 

1.5798 

6577 

1.5204 

6830  1.4641 

40 

21 

5855 

1.7079 

6092 

1.6415 

6334 

1.5788 

6581 

1.5195 

6834  1.4632 

39 

22 

5859 

1.7067 

6096 

1.6404 

6338 

1.5778 

6585 

1.5185 

6839  1.4623 

38 

23 

5863 

1.7056 

6100 

1.6393 

6342 

1.5768 

6590 

1.5175 

6843  1.4614 

37 

24 

5867 

1.7045 

6104 

1.6383 

6346 

1.5757  . 

6594 

1.5166 

6847  1.4605 

36 

25 

5871 

1.7033 

6108 

1.6372 

6350 

1.5747 

6598 

1.5156 

6851  1.4596 

35 

26 

5875 

1.7022 

6112 

1.6361 

6354 

1.5737 

6602 

1.5147 

6856  1.4586 

34 

27 

5879 

1.7011 

6116 

1.6351 

6358 

1.5727 

6606 

1.5137 

6860  1.4577 

33 

28 

5883 

1.6999 

6120 

1.6340 

6363 

1.5717 

6610 

1.5127 

6864  1.4568 

32 

29 

5887 

1.6988 

6124 

1.6329 

6367 

1.5707 

6615 

1.5118 

6869  1.4559 

31 

30 

5890 

1.6977 

6128 

1.6319 

6371 

1.5697 

6619 

1.5108 

6873  1.4550 

30 

31 

5894 

1.6965 

6132 

1.6308 

6375 

1.5687 

6623 

1.5099  - 

6877  1.4541 

29 

32 

5898 

1.6954 

6136 

1.6297 

6379 

1.5677 

6627 

1.5089 

6881  1.4532 

28 

33 

5902 

1.6943 

6140 

1.6287 

6383 

1.5667 

6631 

1.5080 

6886  1.4523 

27 

34 

5906 

1.6932 

6144 

1.6276 

6387 

1.5657 

6636 

1.5070 

6890  1.4514 

26 

35 

5910 

1.6920 

6148 

1.6265 

6391 

1.5647 

6640 

1.5061 

6894  1.4505 

25 

36 

5914 

1.6909 

6152 

16255 

6395 

1.5637 

6644 

1.5051 

6899  1.4496 

24 

37 

5918 

1.6898 

6156 

1.6244 

6399 

1.5627 

6648 

1.5042 

6903  1.4487 

23 

38 

5922 

1.6887 

6160 

1.6234 

6403 

1.5617 

6652 

1.5032 

6907  1.4478 

22 

39 

5926 

1.6875 

6164 

1.6223 

6408 

1.5607 

6657 

1.5023 

6911  1.4469 

21 

40 

5930 

1.6864 

6168 

1.6212 

6412 

1.5597 

6661 

1.5013 

6916  1.4460 

20 

41 

5934 

1.6853 

6172 

1.6202 

6416 

1.5587 

6665 

1.5004 

6920  1.4451 

19 

42 

5938 

1.6842 

6176 

1.6191 

6420 

1.5577 

6669 

1.4994 

6924  1.4442 

18 

43 

5942 

1.6831 

6180 

1.6181 

6424 

1.5567 

6673 

1.4985 

6929  1.4433 

17 

44 

5945 

1.6820 

6184 

1.6170 

6428 

1.5557 

6678 

1.4975 

6933  1.4424 

16 

45 

5949 

1.6808 

6188 

1.6160 

6432 

1.5547 

6682 

1.4966 

6937  1.4415 

15 

46 

5953 

1.6797 

6192 

1.6149 

6436 

1.5537 

6686 

1.4957 

6942  1.4406 

14 

47 

5957 

1.6786 

6196 

1.6139 

6440 

1.5527 

6690 

1.4947 

6946  1.4397 

13 

48 

5961 

1.6775 

6200 

1.6128 

6445 

1.5517 

6694 

1.4938 

6950  1.4388 

12 

49 

5965 

1.6764 

6204 

1.6118 

6449 

1.5507 

6699 

1.4928 

6954  1.4379 

11 

50 

5969 

1.6753 

6208 

1.61,07 

6453 

1.5497 

6703 

1.4919 

6959  1.4370 

IO 

51 

5973 

1.6742 

6212 

1.6097 

6457 

1.5487 

6707 

1.4910 

6963  1.4361 

9 

52 

5977 

1.6731 

6216 

1.6087 

6461 

1.5477 

6711 

1.4900 

6967  1.4352 

8 

53 

5981 

1.6720 

•  6220 

1.6076 

6465 

1.5468 

6716 

1.4891 

6972  1.4344 

7 

54 

5985 

1.6709 

6224 

1.6066 

6469 

1.5458 

6720 

1.4882 

,£976  1.4335 

6 

55 

5989 

1.6698 

6228 

1.6055 

6473 

1.5448 

6724 

1.4872 

6980  1.4326 

5 

56 

5993 

16687 

6233 

1.6045 

6478 

1.5438 

6728 

1.4863 

6985  1.4317 

4 

57 

5997 

1.6676 

6237 

1.6034 

6482 

1.5428 

6732 

1.4854 

6989  1.4308 

3 

58 

6001 

1.6665 

6241 

1.6024 

6486 

1.5418 

6737 

1.4844 

6993  1.4299 

2 

59 

6005 

1.6654 

6245 

1.6014 

6490 

1.5408 

6741 

1.4835 

6998  1.4290 

1 

60 

6009 

1.6643 

6249 

1.6003 

6494 

1.5399 

6745 

1.4826 

7002  1.4281 

O 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot   tan 

f 

59° 

58° 

57° 

56° 

55° 

/ 

66 

NATTJBAL  TANGENTS  AND 

COTANGENTS. 

/ 

35° 

36° 

37° 

38° 

39° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

7002 

1.4281 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

60 

1 

7006 

1.4273 

7270 

1.3755 

7540 

1.3262 

7818 

1.2792 

8103 

1.2342 

59 

2 

7011 

1.4264 

7274 

1.3747 

7545 

1.3254 

7822 

1.2784 

8107 

1.2334 

58 

3 

7015 

1.4255 

7279 

1.3739 

7549 

1.3246 

7827 

1.2776 

8112 

1.2327 

57 

4 

7019 

1.4246 

7283 

1.3730 

7554 

1.3238 

7832 

1.2769 

8117 

1.2320 

56 

5 

7024 

1.4237 

7288 

1.3722 

7558 

1.3230 

7836 

1.2761 

8122 

1.2312 

55 

6 

7028 

1.4229 

7292 

1.3713 

7563 

1.3222 

7841 

1.2753 

8127 

1.2305 

54 

7 

7032 

1.4220 

7297 

1.3705 

7568 

1.3214 

7846 

1.2746 

8132 

1.2298 

53 

8 

7037 

1.4211 

7301 

1.3697 

7572 

1.3206 

7850 

1.2738 

8136 

1.2290 

52 

9 

7041 

1.4202 

7306 

1.3688 

7577 

1.3198 

7855 

1.2731 

8141 

1.2283 

51 

io 

7046 

1.4193 

7310 

1.3680 

7581 

1.3190 

7860 

1.2723 

8146 

1.2276 

50 

11 

7050 

1.4185 

7314 

1.3672 

7586 

1.3182 

7865 

1.2715 

8151 

1.2268 

49 

12 

7054 

1.4176 

7319 

1.3663 

7590 

1.3175 

7869 

1.2708 

8156 

1.2261 

48 

13 

7059 

1.4167 

7323 

1.3655 

7595 

1.3167 

7874 

1.2700 

8161 

1.2254 

47 

14 

7063 

1.4158 

7328 

1.3647 

7600 

1.3159 

7879 

1.2693 

8165 

1.2247 

46 

15 

7067 

1.4150 

7332 

1.3638 

7604 

1.3151 

7883 

1.2685 

8170 

1.2239 

45 

16 

7072 

1.4141 

7337 

1.3630 

7609 

1.3143 

7888 

1.2677 

8175 

1.2232 

44 

17 

7076 

1.4132 

7341 

1.3622 

7613 

1.3135 

7893 

1.2670 

8180 

1.2225 

43 

18 

7080 

1.4124 

7346 

1.3613 

7618 

1.3127 

7898 

1.2662 

8185 

1.2218 

42 

19 

7085 

1.4115 

7350 

1.3605 

7623 

1.3119 

7902 

1.2655 

8190 

1.2210 

41 

20 

7089 

1.4106 

7355 

1.3597 

7627 

1.3111 

7907 

1.2647 

8195 

1.2203 

40 

21 

7094 

1.4097 

7359 

1.3588 

7632 

1.3103 

7912 

1.2640 

8199 

1.2196 

39 

22 

7098 

1.4089 

7364 

1.3580 

7636 

1.3095 

7916 

1.2632 

8204 

1.2189 

38 

23 

7102 

1.4080 

7368 

1.3572 

7641 

1.3087 

7921 

1.2624 

8209 

1.2181 

37 

24 

7107 

1.4071 

7373 

1.3564 

7646 

1.3079 

7926 

1.2617 

8214 

1.2174 

36 

25 

7111 

1.4063 

7377 

1.3555 

7650 

1.3072 

7931 

1.2609 

8219 

1.2167 

35 

26 

7115 

1.4054 

7382 

1.3547 

7655 

1.3064 

7935 

1.2602 

8224 

1.2160 

34 

27 

7120 

1.4045 

7386 

1.3539 

7659 

1.3056 

7940 

1.2594 

8229 

1.2153 

33 

28 

7124 

1.4037 

7391 

1.3531 

7664 

1.3048 

7945 

1.2587 

8234 

1.2145 

32 

29 

7129 

1.4028 

7395 

1.3522 

7669 

1.3040 

7950 

1.2579  • 

8238 

1.2138 

31 

30 

7133 

1.4019 

7400 

1.3514 

7673 

1.3032 

7954 

1.2572 

8243 

1.2131 

30 

31 

7137 

1.4011 

7404 

1.3506 

7678 

1.3024 

7959 

1.2564 

8248 

1.2124 

29 

32 

7142 

1.4002 

7409 

1.3498 

7683 

1.3017 

7964 

1.2557 

8253 

1.2117 

28 

33 

7146 

1.3994 

7413 

1.3490 

7687 

1.3009 

7969 

1.2549 

8258 

1.2109 

27 

34 

7151 

1.3985 

7418 

1.3481 

7692 

1.3001 

7973 

1.2542 

8263 

1.2102 

26 

35 

7155 

1.3976 

7422 

1.3473 

7696 

1.2993 

7978 

1.2534 

8268 

1.2095 

25 

36 

7159 

1.3968 

7427 

1.3465 

7701 

1.2985 

7983 

1.2527 

8273 

1.2088 

24 

37 

7164 

1.3959 

7431 

1.3457 

7706 

1.2977 

7988 

1.2519 

8278 

1.2081 

23 

38 

7168 

1.3951 

7436 

1.3449 

7710 

1.2970 

7992 

1.2512 

8283 

1.2074 

22 

39 

7173 

1.3942 

7440 

1.3440 

7715 

1.2962 

7997 

1.2504 

8287 

1.2066 

21 

40 

7177 

1.3934 

7445 

1.3432 

7720 

1.2954 

8002 

1.2497 

8292 

1.2059 

20 

41 

7181 

1.3925 

7449 

1.3424 

7724 

1.2946 

8007 

1.2489 

8297 

1.2052 

19 

42 

7186 

1.3916 

7454 

1.3416 

7729 

1.2938 

8012 

1.2482 

8302 

1.2045 

18 

43 

7190 

1.3908 

7458 

1.3408 

7734 

1.2931 

8016 

1.2475 

8307 

1.2038 

17 

44 

7195 

1.3899 

7463 

1.3400 

7738 

1.2923 

8021 

1.2467 

8312 

1.2031 

16 

45 

7199 

1.3891 

7467 

1.3392 

7743 

1.2915 

8026 

1.2460 

8317 

1.2024 

15 

46 

7203 

1.3882 

7472 

1.3384 

7747 

1.2907 

8031 

1.2452 

8322 

1.2017 

14 

47 

7208 

1.3874 

7476 

1.3375 

7752 

1.2900 

8035 

1.2445 

8327 

1.2009 

13 

48 

7212 

1.3865 

7481 

1.3367 

7757 

1.2892 

8040 

1.2437 

8332 

1.2002 

12 

49 

7217 

1.3857 

7485 

1.3359 

7761 

1.2884 

8045 

1.2430 

8337 

1.1995 

11 

50 

7221 

1.3848 

7490 

1.3351 

7766 

1.2876 

8050 

1.2423 

8342 

1.1988 

IO 

51 

7226 

1.3840 

7495 

1.3343 

7771 

1.2869 

8055 

1.2415 

8346 

1.1981 

9 

52 

7230 

1.3831 

7499 

1.3335 

7775 

1.2861 

8059 

1.2408 

8351 

1.1974 

8 

53 

7234 

1.3823 

7504 

1.3327 

7780 

1.2853 

8064 

1.2401 

8356 

1.1967 

7 

54 

7239 

1.3814 

750S 

1.3319 

7785 

1.2846 

8069 

1.2393 

8361 

1.1960 

6 

55 

7243 

1.3806 

7513 

1.3311 

7789 

1.2838 

8074 

1.2386 

8366 

1.1953 

5 

56 

7248 

1.3798 

7517 

1.3303 

7794 

1.2830 

8079 

1.2378 

8371 

1.1946 

4 

57 

7252 

1.3789 

7522 

1.3295 

7799 

1.2822 

8083 

1.2371 

8376 

1.1939 

3 

58 

7257 

1.3781 

7526 

1.3287 

7803 

1.2815 

8088 

1.2364 

8381 

1.1932 

2 

59 

7261 

1.3772 

7531 

1.3278 

7808 

1.2807 

8093 

1.2356 

8386 

1.1925 

1 

60 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

8391 

1.1918 

O 

f 

cot   tan 

54° 

cot   tan 
53° 

cot   tan 

52° 

cot   tan 
51° 

cot   tan 
50° 

t 

NATUBAL  TANGENTS   AND   COTANGENTS.  67 


t 

40° 

41° 

42° 

43° 

44° 

r 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

8391 

1.1918 

8693 

1.1504 

9004 

1.1106 

9325 

1.0724 

9657 

1.0355 

60 

1 

8396 

1.1910 

8698 

1.1497 

9009 

1.1100 

9331 

1.0717 

9663 

1.0349 

59 

2 

8401 

1.1903 

8703 

1.1490 

9015 

1.1093 

9336 

1.0711 

9668 

1.0343 

58 

3 

8406 

1.1896 

8708 

1.1483 

9020 

1.1087 

9341 

1.0705 

9674 

1.0337 

57 

4 

8411 

1.1889 

8713 

1.1477 

9025 

1.1080 

9347 

1.0699 

9679 

1.0331 

56 

5 

8416 

1.1882 

8718 

1.1470 

9030 

1.1074 

9352 

1.0692 

9685 

1.0325 

55 

6 

8421 

1.1875 

8724 

1.1463 

9036 

1.1067 

9358 

1.0686 

9691 

1.0319 

54 

7 

8426 

1.1868 

8729 

1.1456 

9041 

1.1061 

9363 

1.0680 

9696 

1.0313 

53 

8 

8431 

1.1861 

8734 

1.1450 

9046 

1.1054 

9369 

1.0674 

9702 

1.0307 

52 

9 

8436 

1.1854 

8739 

1.1443 

9052 

1.1048 

9374 

1.0668 

9708 

1.0301 

51 

io 

8441 

1.1847 

8744 

1.1436 

9057 

1.1041 

9380 

1.0661 

9713 

1.0295 

50 

11 

8446 

1.1840 

8749 

1.1430 

9062 

1.1035 

9385 

1.0655 

9719 

1.0289 

49 

12 

8451 

1.1833 

8754 

1.1423 

9067 

1.1028 

9391 

1.0649 

9725 

1.0283 

48 

13 

8456 

1.1826 

8759 

1.1416 

9073 

1.1022 

9396 

1.0643 

9730 

1.0277 

47 

14 

8461 

1.1819 

8765 

1.1410 

9078 

1.1016 

9402 

1.0637 

9736 

1.0271 

46 

15 

8466 

1.1812 

8770 

1.1403 

9083 

1.1009 

9407 

1.0630 

9742 

1.0265 

45 

16 

8471 

1.1806 

8775 

1.1396 

9089 

1.1003 

9413 

1.0624 

9747 

1.0259 

44 

17 

8476 

1.1799 

8780 

1.1389 

9094 

1.0996 

9418 

1.0618 

9753 

1.0253 

43 

18 

8481 

1.1792 

8785 

1.1383 

9099 

1.0990 

9424 

1.0612 

9759 

1.0247 

42 

19 

8486 

1.1785 

8790 

1.1376 

9105 

1.0983 

9429 

1.0606 

9764 

1.0241 

41 

|20 

8491 

1.1778 

8796 

1.1369 

9110 

1.0977 

9435 

1.0599 

9770 

1.0235 

40 

\  21 

8496 

1.1771 

8801 

1.1363 

9115 

1.0971 

9440 

1.0593 

9776 

1.0230 

39 

2z 

8501 

1.1764 

8806 

1.1356 

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